Binary Codes from Reflexive Uniform Subset Graphs on 3-sets *

We examine the binary codes C2(Ai + I) from matrices Ai + I where Ai is an adjacency matrix of a uniform subset graph Γ(n, 3, i) of 3-subsets of a set of size n with adjacency defined by subsets meeting in i elements of Ω, where 0 ≤ i ≤ 2. Most of the main parameters are obtained; the hulls, the duals, and other subcodes of the C2(Ai + I) are also examined. We obtain partial PD-sets for some of the codes, for permutation decoding.


Introduction
Codes from the row span over finite fields of incidence matrices of regular graphs have been shown to have uniform properties that can result in the graphs being retrieved from the code: see [?, ?] for the general results concerning these codes and for references to previous work on various classes of graphs that led to formulation of the general result.In contrast, codes from adjacency matrices for graphs have been found to have no uniform properties in general, and the various classes appear to need to be examined separately, although similar techniques can be used over various classes.In particular, we have observed that for uniform subset graphs Γ(n, k, r) = (V, E) where the vertices V are k-subsets of a set of size n, with adjacency defined by the k-subsets meeting in r points, the codes from the adjacency matrix over any field are intimately related to a set of k codes on V , denoted by W i for 1 ≤ i ≤ k − 1, and W Π ⊆ W ⊥ i , that are defined independently of the actual graph, and that can be studied separately.
In this paper we examine binary codes from adjacency matrices from the uniform subset graphs on 3-sets: let Γ i n = Γ(n, 3, i) = (V, E i ) denote the uniform subset graph with V the set of 3-subsets of Ω = {1, . . ., n}, for i = 0, 1, 2, where adjacency in Γ i n is defined by vertices being adjacent if the 3-subsets meet in i points.The binary codes C 2 (A i ) from the row span over F 2 of adjacency matrices A i for these graphs were examined in [?], and the ternary codes in [?].We look here at the binary codes C 2 (A i + I) from the matrices A i + I, these being the adjacency matrices of the reflexive graph RΓ i n , which is obtained from Γ i n by including a loop at every vertex.The binary codes from A i + I have similarities with those from A i (for example, C 2 (A i ) ⊥ ⊆ C 2 (A i + I)) and results from [?] can be used to establish some results here.However, there are major differences that make these codes worthy of study, in particular that all of the codes have minimum weight at least n−2, whereas some of the C 2 (A i ) are the full space F

|V |
2 .This also applies to some other graphs: see [?].In a separate paper [?] we have similar results for the ternary codes, following the work of [?].
We summarize our main results concerning the codes C 2 (A i + I) from the row span over F 2 of A i + I below.The row of A i + I corresponding to the vertex x is denoted by s i x , and the row of A i corresponding to the vertex x by r i x , so s i x = r i x + v x .Other notation used can be found in the following sections, including the codes W 1 , W 2 , W Π mentioned above.A summary of these results can be found in Table ?? at the end of the paper.
Theorem 1.For n ≥ 7 let A i be an adjacency matrix for the uniform subset graph Γ i n on 3-sets, C i = C 2 (A i +I), for i = 0, 1, 2.
1.For n odd, C 0 is a [ n 3 , n 2 , n − 2] 2 code; for n ≥ 9 the minimum words are the w a,b = c∈Ω\{a,b} v {a,b,c} for a, b ∈ Ω.
The minimum weight of C ⊥ 0 is 8.
2. C 1 is a [ n 3 , n, n−1 2 ] 2 code, and C 1 = w a | a ∈ Ω , where w a = b,c∈Ω\{a} v {a,b,c} .It has weight distribution given by n r words of weight n r = r n−r 2 + r 3 for each 0 ≤ r ≤ n.For n ≥ 8, the minimum words are the w a .For n = 7 there are a further 21 words from r = 5.
The minimum weight of C ⊥ 1 is 4.
For n odd, C 2 is a [ n 3 , n−1 2 , n − 2] 2 code with a basis of the minimum words w a,b .The minimum weight of C ⊥ 2 is 4.
The symmetric group S n acts transitively of degree n 3 as a permutation group on each of these codes.The proof of these results, together with further results regarding the duals, hulls, the codes generated by the difference of two rows of A i + I, and inter-relationships amongst the codes, can be found in the sections to follow, which are preceeded by a short section giving background definitions and terminology.
In the final section we define the codes W i and W Π for general uniform subset graphs Γ(n, k, r).

Terminology and background
The notation for designs and codes is as in [?].An incidence structure D = (P, B, J ), with point set P, block set B and incidence J is a t-(v, k, λ) design if |P| = v, every block B ∈ B is incident with precisely k points, and every t distinct points are together incident with precisely λ blocks.The design is symmetric if it has the same number of points and blocks.The code C F (D) of the design D over the finite field F is the space spanned by the incidence vectors of the blocks over F .If Q is any subset of P, then we will denote the incidence vector of Q by v Q , and if Q = {P } where P ∈ P, then we will write and is a subspace of F P , the full vector space of functions from P to F .For any w ∈ F P and P ∈ P, w(P ) denotes the value of w at P .If F = F p then the p-rank of the design, written rank p (D), is the dimension of its code C F (D); for F = F p we usually write C p (D) for C F (D).
All the codes here are linear codes, and the notation [n, k, d] q will be used for a q-ary code C of length n, dimension k, and minimum weight d, where the weight wt(v) of a vector v is the number of non-zero coordinate entries.Vectors in a code are also called words.The support, Supp(v), of a vector v is the set of coordinate positions where the entry in v is non-zero.So |Supp(v)| = wt(v).The distance d(u, v) between two vectors u, v is the number of coordinate positions in which they differ, i.e., wt(u − v).A generator matrix for C is a k × n matrix made up of a basis for C, and the dual code C ⊥ is the orthogonal under the standard inner product (, ), i.e.C ⊥ = {v ∈ F n | (v, c) = 0 for all c ∈ C}.The hull of a code C is the self-orthogonal code C ∩ C ⊥ .A check matrix for C is a generator matrix for C ⊥ .The all-one vector will be denoted by , and is the vector with all entries equal to 1.If we need to specify the length m of the all-one vector, we write  m .We call two linear codes isomorphic (or permutation isomorphic) if they can be obtained from one another by permuting the coordinate positions.An automorphism of a code C is an isomorphism from C to C. The automorphism group will be denoted by Aut(C), also called the permutation group of C, and denoted by PAut(C) in [?].Any code is isomorphic to a code with generator matrix in so-called standard form, i.e. the form [I k | A]; a check matrix then is given by [−A T | I n−k ].The set of the first k coordinates in the standard form is called an information set for the code, and the set of the last n − k coordinates is the corresponding check set.
The graphs, Γ = (V, E) with vertex set V and edge set E, discussed here are undirected with no loops, apart from the case where all loops are included, in which case the graph is called reflexive.If x, y ∈ V and x and y are adjacent, we write x ∼ y, and xy or [x, y] for the edge in E that they define.The set of neighbours of x ∈ V is denoted by N (x), and the valency of x is |N (x)|.Γ is regular if all the vertices have the same valency.A path of length r from vertex x to vertex y is a sequence x i , for 0 ≤ i ≤ r − 1, of distinct vertices with x = x 0 , y = x r−1 , and in which case we write it (x 0 , . . ., x r−1 ).The graph is connected if there is a path between any two vertices; d(x, y) denotes the length of the shortest path from x to y.
An adjacency matrix A is a |V | × |V | symmetric matrix with entries a ij such that a ij = 1 if vertices x i and x j are adjacent, and a ij = 0 otherwise.The neighbourhood design of Γ is the symmetric 1-(|V |, k, k) design formed by taking the points to be the vertices of the graph and the blocks to be the sets of neighbours of a vertex, for each vertex, i.e. an adjacency matrix as an incidence matrix for the design.If Γ = (V, E) is a graph with adjacency matrix A then A + I |V | is an adjacency matrix for the reflexive graph from Γ.
The code of Γ over a finite field F is the row span of an adjacency matrix A over the field F , denoted by C F (Γ) or C F (A).The dimension of the code is the rank of the matrix over F , also written rank p (A) if F = F p , in which case we will speak of the p-rank of A or Γ, and write C p (Γ) or C p (A) for the code.It is also the code over F p of the neighbourhood design.
The uniform subset graph Γ(n, k, r) has for vertices the set of all subsets of size k of a set of size n with two k-subsets x and y defined to be adjacent if |x ∩ y| = r.The symmetric group S n always acts on Γ(n, k, r), transitively on vertices and edges.
Permutation decoding was first developed by MacWilliams [?] and involves finding a set of automorphisms of a code called a PD-set.The method is described fully in MacWilliams and Sloane [?,Chapter 16,p. 513] and Huffman [?, Section 8].In [?] and [?] the definition of PD-sets was extended to that of s-PD-sets for s-error-correction: Definition 1.If C is a t-error-correcting code with information set I and check set C, then a PD-set for C is a set S of automorphisms of C which is such that every t-set of coordinate positions is moved by at least one member of S into the check positions C.
For s ≤ t an s-PD-set is a set S of automorphisms of C which is such that every s-set of coordinate positions is moved by at least one member of S into C.
The algorithm for permutation decoding is as follows: we have a t-error-correcting [n, k, d] q code C with check matrix H in standard form.Thus the generator matrix G = [I k |A] and H = [−A T |I n−k ], for some A, and the first k coordinate positions correspond to the information symbols.Any vector v of length k is encoded as vG.Suppose x is sent and y is received and at most t errors occur.Let S = {g 1 , . . ., g s } be the PD-set.Compute the syndromes H(yg i ) T for i = 1, . . ., s until an i is found such that the weight of this vector is t or less.Compute the codeword c that has the same information symbols as yg i and decode y as cg −1 i .Notice that this algorithm actually uses the PD-set as a sequence.Thus it is expedient to index the elements of the set S by the set {1, 2, . . ., |S|} so that elements that will correct a small number of errors occur first.Thus if nested s-PD-sets are found for all 1 < s ≤ t then we can order S as follows: find an s-PD-set S s for each 0 ≤ s ≤ t such that S 0 ⊂ S 1 . . .⊂ S t and arrange the PD-set S as a sequence in this order: S = [S 0 , (S 1 − S 0 ), (S 2 − S 1 ), . . ., (S t − S t−1 )].
(Usually one takes S 0 = {id}.) There is a bound on the minimum size that a PD-set S may have, due to Gordon [?], from a formula due to Schönheim [?], and quoted and proved in [?]: This result can be adapted to s-PD-sets for s ≤ t by replacing t by s in the formula.
3 The graphs Γ i n All codes here are linear and binary, with all spans being over the field F 2 .
In this section we establish some general relationships amongst the codes from the matrices A i + I.
) denote the uniform subset graph with V the set of 3-subsets of Ω = {1, . . ., n}, with i = 0, 1, 2, and A i an adjacency matrix.Consider the the code C 2 (A i + I) = C Ai+I for i = 0, 1, 2 (in the notation of [?]).We use the results of [?] concerning the codes C 2 (A i ) = C Ai .We denote the row of A i + I corresponding to the vertex x by s i x , and the row of A i corresponding to the vertex x by r i x , so s i x = r i x + v x .The neighbours of x in Γ i n are denoted by N i (x).This graph that includes a loop at each vertex is called a reflexive graph and we will denote it here by RΓ i n .From [?, Proposition 2.2], for each of the i, and some of the properties of the C A i+I can be deduced from results in [?]; we will use those results when we can.
Notation: In the following we will write Also from [?, Lemma 2.2],  ∈ C i for each i, and it is clear that Write x i ∼ y for x adjacent to y in Γ i n , i = 0, 1, 2. The valency ν i for Γ i n is given by: The neighbourhood designs of these three graphs and their reflexive associates, respectively, will be denoted by , respectively.The automorphism groups of the graphs, designs and codes always contains S n , but may be larger in some cases: see Examples ??, ??.
Using design terminology, we may refer to the vertices x as points and we will denote the block of the design R i n determined by x = {a, b, c} and its neighbours by: so that xi = Supp(s i x ).The codes W 1 , W 2 and W Π defined below have a role to play in the codes from A i and those from A i + I.In fact for uniform subset graphs on k-sets, similarly defined codes W i for 1 ≤ i ≤ k − 1 arise: see [?].In addition, these codes can be defined over any field F p and will be similarly related to the codes from the uniform subset graphs.In [?, ?] they are defined as ternary codes.
From [?, Lemma 4]: 2 , and dim(W 1 ) = n.Lemma 1.For all n ≥ 7, W 2 has dimension n−1 2 and, for any fixed a ∈ Ω, Proof: For any n,  = i,j w i,j , so  ∈ W 2 .Also and if n is even then w 1,n is a sum of an even number of w 1,i , so W 2 = W * 2 .To show that {w a,b | a, b = n} is a linearly independent set, suppose w = a,b∈Ω\{n} α a,b w a,b = 0. Then the coordinate entry at {a, b, n} is α a,b , so α a,b = 0 for all a, b, and the n−1 2 generators are linearly independent.If n is odd, then wt(w a,b ) = n − 2 is odd, so (w a,b , ) = 1 for all a, b, and thus  ∈ (W * 2 ) ⊥ , but  ∈ (W 2 ) ⊥ .That the minumum weight is 2n − 6 follows from [?], Proposition 6, and the proof of that proposition, and more, in Section 4 of that paper, since the proof holds for any prime p. Then computations with Magma deal with n ≤ 23.
The following definition is given in a way that can apply to the codes over any characteristic.The word w π is the word w(π) of [?, Equation 8].
and X c the set of their complements in ∆.Define the weight-8 vector and and w({x, y, n}) = α x,y + β x + β y = 0.This gives α x,z + α y,z + β z = 0, and thus α y,z = β x , from which it follows that α x,y = β z = c, a constant, for all x, y, z and thus the words in S are linearly independent.Thus 3c = 0 and so c = 0. Thus dim(W ⊥ Π ) ≥ |S| = n 2 and the result follows.
Note: 1.A basis for W Π is thus given in [?, Lemma 10], since a linearly independent set is given there.It follows that the result also shows an information set for W Π .This shows that the dimension of W Π over any field F p with p = 3, is as given, since the proof that dim(W Π ) ≥ n 3 − n 2 does not depend on the characteristic of the field.For p = 3 equality is proved separately in [?].
Notice that the E i are all even-weight codes.
Proof: For n ≡ 3 (mod 4), all the ν i + 1 and |V | are odd, so  is the sum of all the n 3 s i x , which is an odd number, so  ∈ E i for any i in this case.
For n ≡ 0 (mod 4), ν i + 1 is odd for i = 0, 1, and |V | is even, so  is the sum of all the rows in each case, and this is an even number, so For n ≡ 1 (mod 4), ν i + 1 is odd for i = 0, 2, and |V | is even, so  is the sum of all the rows in each case, and this is an even number, so  ∈ E i for i = 0, 2. Also,  = a∈Ω w a (by Proposition ??) so it is a sum of an odd number of w a 's and thus from the proof of that proposition, the sum of an odd number of s 1 x 's, and thus  ∈ E 1 .
For n ≡ 2 (mod 4), from Lemma ??, for all i since all the ν i + 1 are even, we see that (s 1 x , s 0 y ) = (, s 0 y ) + (u, s 0 y ) ≡ 0 (mod 2) which is a contradiction since this is ≡ 1 (mod 2) by Lemma ?? (3).Thus  ∈ E 1 , and a similar argument gives  ∈ E 2 .Thus also  ∈ E 0 .Now we look at the containments amongst the C i , using ideas and methods from [?], but also previous work from [?] on the binary codes from the adjacency matrices of the Johnson graphs.
In [?], for each x ∈ V , the word w x = y i ∼x r y was defined and used to find relations amongst the C Ai .Here we define, for x ∈ V , where we add the superscript i to indicate the various Γ i n .We can use the results in [?] to obtain the following table for the w i x , for any fixed x ∈ V , recalling that for any vector w, w(y) denotes it value at the coordinate position y.
x , it is clear that w ∈ C ⊥ 0 and we have equality.
Lemma 8.For all n ≥ 7, for all i = 0, 1, 2, j = 1, 2, C ⊥ i , W ⊥ j have minimum weight at least 4, and exactly 4 for i, j = 1, 2. C ⊥ 0 has minimum weight at most 8.Further 1. C ⊥ 1 has words of weight 4 with support of each of the forms all of which meet a weight-8 word w π in four points.The set of supports of such words form the blocks of a 1-( n 3 , 6, 3(n − 3)(n − 4)) design such that two points of V are together on 0, 4, or 3(n − 4) blocks.These weight-6 words are not in C ⊥ 0 .For n = 8 these are the only words of weight 6 in C ⊥ 2 .
Proof: Since W Π ⊆ C ⊥ i for all i, the minimum weight of any of the C ⊥ i is at most 8. Further, since  ∈ C i for all the i, C ⊥ i is an even-weight code, and a simple argument eliminates the possibility of a weight-2 word.It can be verified directly that the words with supports as shown are in the duals as asserted, and that the three types are the only weight-4 words in C ⊥ 1 follows by simply considering the possibilities.That the words with support of the form Equation (??) are the only weight-4 words in C ⊥ 2 follows from an easy argument.For the words of weight 6 of the form Equation (??) in C ⊥ 2 , each 5-set from Ω gives 5 2 = 10 words, so there are 10 n 5 in all.This gives r = 3(n − 3)(n − 4) for the replication number.If x, y ∈ V have |x ∩ y| = 0, they will be on no such blocks together; if |x ∩ y| = 1 they will be on four such blocks (from the 5-set x ∪ y); if |x ∩ y| = 2 then they will be on 3(n − 4) blocks.
For other words of weight 6 in C ⊥ 2 , if n is odd then we can argue as in the proof of [?, Proposition 3] that these are the only words of weight 6 in C ⊥ 2 .For n even, if n = 8 then w a,b ∈ C ⊥ 2 and has weight 6, but for n = 10, 12, Magma shows that these are the only words of weight 6 in C ⊥ 2 .For n ≥ 14 we argue as follows: let Notice that if we suppose that w is not of the form of Equation (??) then w can meet any weight-4 vector u in C ⊥ 2 in at most two points, since if it met in three points then w + u would be a weight-4 vector and hence of the form Equation (??), showing that w is of the form of Equation (??).Since |Λ| ≤ 18, if n > 18 there is such an a ∈ Ω so we have the result.If n = 18 it is clear that Λ = Ω is impossible, and for n = 16, 14 the argument is similar.We can use Magma for n = 12, 10.
The following lemma is proved for the ternary codes in [?, Propositions 4,5], and the result holds, with the proof virtually the same, for the binary case.Thus we omit the proof.The minimum weight of C i is at least n − 2 for i = 0, 1, 2 and all n ≥ 7.
Note: For n = 7, the dual of the code spanned by the weight-8 vectors in Lemma ?? has words of weight n − 2 = 5 other than the w a,b ; this code is C 0 and for n = 7, wt(s 0 x ) = 5.
The following mod 2 values of the inner products of the rows of the A i + I can be verified directly.In each case, the identity holds for all x, y ∈ V .
We can use Lemma ?? to obtain the following lemma: Lemma 12.For all n ≥ 7, E i = C Ai+1 , and Proof: For the first statement, note that E i is an even weight code since wt(s i x + s i y ) = 2wt(s i x ) − 2|s i x ∩ s i y |.Thus if ν i + 1 is odd, the claim is immediately true.This covers n ≡ 3 (mod 4) for then all the ν i + 1 are odd.
The other cases can be deduced from the list above.Thus if E 1 = C 1 then, since s 1 x + s 1 y ∈ C ⊥ 2 , we would have also s 1 x ∈ C ⊥ 2 , which is false.Similarly E 2 = C 2 .The same follows for E 0 for n even, so we are left with n ≡ 1 (mod 4).But in this case ν 0 + 1 is odd, so the argument for n ≡ 3 (mod 4) holds here for E 0 .
Proof: First notice that if A 2 = I then (A + I)(A + I) = 0, so C A+I is self-orthogonal.We will show that (r i x , r i y ) = δ x,y for i = 0, 1 when n ≡ 2 (mod 4) and for i = 2 and any even n.

The codes C 1
We look first at the codes C 1 since more can be proved about this case than about C 0 or C 2 .
Its weight distribution is given by n r words of weight n r = r n−r 2 + r 3 from the sum of r distinct w a , for each 0 ≤ r ≤ n.For n ≥ 8, the minimum words are the w a .For n = 7 there are another 21 words from r = 5.C ⊥ 1 has minimum weight 4.
For n ≡ 0 (mod 4), Proof: We first note that it is easy to prove that, for x = {a, b, c} ∈ V , and for all n, and hence that w a = s 1 {a,b,c} + s 1 {a,b,d} + s 1 {a,c,d} for any distinct a, b, c, d ∈ Ω.Thus C 1 = W 1 , so dim(C 1 ) = n.It is also easy to see that the set of n rows s 1 x with x ∈ I can be put into row echelon form.For the weight distribution we follow the same reasoning as in [?, Lemma 6].Thus let ∆ = {a 1 , . . ., a r } ⊆ Ω, where 0 ≤ r ≤ n, and let 3 .This is for each of the n r choices of ∆.The smallest weight occurs for r = 1, and gives the n words w a of weight n−1 2 : that this is the smallest weight follows because n 1 < n r where r ≥ 2 simplifies to 4r 2 + r(4 − 6n) + (3n 2 − 9n + 6) > 0. The discriminant of this quadratic in r is −12n 2 + 96n − 80 and this is negative for n ≥ 8, so n r > n 1 for all r ≥ 2 for n ≥ 8, and so the w a are all the minimum words.For n = 7, n 5 = n 1 so an additional 21 words occur.For r = n,  = a∈Ω w a .
That the given sets are information sets for C 1 when n ≡ 3 (mod 4) can be verified directly, as can the 2-PD-sets for n = 7 and for n ≥ 8.
Since C ⊥ A1 ⊆ C 1 and for n ≡ 0 (mod 4) they have the same dimension, by [?], they are equal, and equal to C ⊥ A0 , by [?].That C ⊥ 1 has minumum weight 4 was shown in Lemma ??.
Note: 1.The n r are not necessarily distinct for distinct r, but each value of r gives n r of that weight.For example, for n = 7, n 2 = n 6 = 20, and there are 7 2 + 7 6 = 28 words of this weight.However, it can be verified that n r = n s cannot have solutions for n ≡ 0 (mod 4) so in that case there are exactly n distinct non-zero weights.2. Using the Vandermonde identity, it can be verified that n r + n n−r = n 3 , so if n r = n s , then n n−r = n n−s .
The set I \{{n−2, n−1, n}} from Proposition ?? is an information set for E 1 , and the set of automorphisms given there is a nested 2-PD-set of the minimal size for the code.
Proof: Recall that E 1 is an even-weight code spanned by s 1 x + s 1 y and thus also by even sums of the w a .Thus E 1 = W * 1 .So we take r to be even.Note that n 2 = (n − 2)(n − 3).Also we show in Lemma ?? that for all n ≥ 7, For fixed n and r, where n ≥ 7 and n ≥ r ≥ 3 let P (n, r) be the statement n 2 < n r , i.e.
Solving shows this holds for r ≥ 8, so for n, r ≥ 8, P (n, r) is true.We need to consider 3 ≤ r ≤ 7 and for this we consider p(n, r) for these values.Direct computation yields that p(n, 3) > 0 for n ≥ 8; p(n, 4) > 0 for n ≥ 9; p(n, 5) > 0 for n ≥ 9; p(n, 6) > 0 for n ≥ 11; p(n, 7) > 0 for n ≥ 11.Thus we have p(n, r) > 0 for n ≥ 11 and all r ≥ 3, as required.For the 7 ≤ n ≤ 11, direct computation with the n r and the corresponding weights gives the result.
That the given sets are information sets for E 1 can be verified directly, as can the 2-PD-set.
The final statement follows since by [?], for n ≡ 1 (mod 4), C ⊥ A1 is spanned by the w a + w b .
In fact, from [?], we can deduce the following for C 1 and E 1 , giving s-PD-sets for s up to n(n−1) 6 − 1.With notation and information sets as in Propositions ?? and ??: Result 3.For n ≥ 8, taking the following elements of S n in their natural action on triples of elements of Ω = {1, 2, . . ., n}: where Id is the identity element of S n , let 6 − 1, and for E 1 for s ≤ n(n−1) 6 − 1.
Note: 1.This result is stated and proved in [?] for the codes C ⊥ A0 for n ≡ 0 (mod 4) and C ⊥ A1 for n ≡ 1 (mod 4) respectively.These are identified as being the codes spanned by the words w a and w a + w b , respectively, and are thus our codes C 1 and E 1 , respectively, for any n ≥ 7. 2. The sets given in the result are large, and are not nested, but they do correct a lot of errors.Smaller sets can be constructed if s is given some small value, as in Propositions ?? and ?? where s = 2. x + s 1 y | x, y ∈ V = E 1 .For the final statement, note that for n ≡ 3 (mod 4), Hull(C 1 ) = E 1 is spanned by the w a + w b of weight (n − 2)(n − 3) ≡ 0 (mod 4) and since it is self-orthogonal, this proves the statement.
5 The codes C 2 For i = 2, the graph Γ 2 n = J(n, 3), a Johnson graph J(n, k), and the 2-ranks of the adjacency matrices A and A + I are given in [?], while the codes from A are studied in [?].Further results for k = 3 are in [?].We sum up what conclusions we have for the codes C A2+I = C 2 from these results, using in particular [?, Proposition 1].
For any a ∈ Ω, a basis for C 2 is {w b,c | b, c ∈ Ω \ {a}}.
Supposing that r ≤ s ≤ t, then r + s + t ≤ 3t and if r+s 2 (t + l) > 3t we would need more than r + s + t extra elements in S; this is certainly the case if r + s > 6.Thus we need consider the cases where r + s ≤ 6.All the cases give weight at least 3n − 8 with equality only if r = s = t = 0 if n > 8, so that w = s 2 {1,2,3} .For n = 8 there are further words of weight 16 from the difference of two rows of A 2 + I that have n = 8 common non-zero entries, for example s 2 {1,2,3} + s 2 {1,2,4} .We leave the details to the reader.For the basis, we set up an ordering of the elements in V that will be useful when we consider bases for C 0 .Thus we order the set S in some fixed way, and order the rows s 2 z with z ∈ {{1, a, b} | {a, b} ∈ S} in the same way.Now for the columns, the first block of n−2 2 will correspond to the vertices {{n, a, b} | {a, b} ∈ S}.This set of columns is labelled C 1 .The next set of columns, labelled C 2 will correspond to the n−2 2 vertices {{1, a, b} | {a, b} ∈ S}.The next, labelled C 3 will be the n−2 3 vertices of 3-sets on Ω \ {1, n}.Finally we take for Now it is clear from this labelling that the matrix of the s 2 z for z ∈ {{1, a, b} | {a, b} ∈ S} has the identity on the left-hand side, and since n−2 2 is the dimension of C 2 , we do have a basis.That C ⊥ 2 has minimum weight 4 follows from Lemma ??.
Lemma 19.For n ≥ 7, for E 2 : 1. if n is even, the words s 2 x + s 2 y for x 2 ∼ y, have weight 4(n − 4) and there are Proof: 1.For n even, the weight can be checked directly; that there are 3 n 4 of them follows from the observation that for any 4-subset {a, b, c, d} of Ω, 2. For n odd, clearly E 2 = W * 2 and the minimum weight was noted in Lemma ??.That the weight of w a,b + w a,c is as stated can be verified directly.There are 3 n 3 such words as for any 3-subset {a, b, c} of Ω, we get three distinct words.

Note:
The minimum weight of E 2 for n even is 4(n − 4) according to computations with Magma for 8 ≤ n ≤ 14.
6 The codes C 0 From [?, Proposition 1] and [?], we can deduce the following: , which is a basis.For n ≥ 9, the words w a,b are the minimum words.C ⊥ 0 has minimum weight 8 and Proof: When n is odd, C 1 and C 2 are subcodes of C 0 , by Corollary ??, so w i,j , w i ∈ C 0 , by Result ?? and Proposition ??.By Lemma ??, C 0 has minimum weight at least n − 2, but since wt(w a,b ) = n − 2, this is the minimum weight.
That for n > 7 the words w a,b are precisely the minimum words (as they are for C 2 ) follows from Lemma ??.The argument fails for n = 7, when there are other words of weight n − 2 = 5, viz. the s 0 x , as noted earlier.To show that the minimum weight of C ⊥ 0 is 8, we can use the fact that C ⊥ 0 ⊆ C ⊥ 2 , and that the only words of weight 4 and 6 in C ⊥ 2 for n odd have the form from Equation (??) and Equation (??), respectively, and that we have shown that these words are not in z that give a basis for C 2 as in Proposition ??, together with the words Proof: We arrange the vertices in the columns C i as described in Proposition ??.We take the n−2 2 rows that form a basis for C 2 and follow these with the rows showing the words w i , for 1 ≤ i ≤ n − 1.We will not need w n since it can be formed form the w i for 1 ≤ i ≤ n − 1, and  ∈ C 2 .
So we need to show that these words are linearly independent and span C 0 .For this we use the upper n−2 2 rows to reduce the entries for the w i that are in the first n−2 2 columns to zero.For w 1 there are no entries there; all its entries are all of C 2 and all of C 4 .For w i , where 2 ≤ i ≤ n − 1, there are entries at {i, j, n} for j = 1, i, n, i.e. n − 3 entries, so to remove them we add to w i the n − 3 rows s 2 z for z = {1, i, j}.This reduces the entries in C 1 to zero.For the columns in C 2 , the columns corresponding to {1, i, j} will have entries in each of the rows s 2 z taken, i.e. n − 3 entries 1.In the row for w i there are also entries 1 in each of these columns, so these will cancel, leaving 0 here too.For the columns {1, j, k} where j, k = i, there will be precisely two non-zero entries, at the rows corresponding to {1, i, j} and {1, i, k}.When added to w i below, these will not change the zeros already there to 1 s.Thus C 2 is also all zero.For C 3 , again {i, j, k} will occur twice, at {1, i, j} and at {1, i, k}, so these will have no effect on the entries in w i .Finally, for C 4 , the single entry 1 at the column {1, i, n} will change to zero, while the other entries zero will change to 1.
Thus the part of the matrix corresponding the rows w 2 to w n−1 will have zeros up to C 3 where they will correspond to the w i words on Ω \ {1, n}, and are thus still linearly independent, regardless of the remaining part in C 4 .The row for w 1 has zero in C 1 then all entries in C 2 are 1.Thus it cannot be dependent on the upper words or the w i for 2 ≤ i ≤ n − 1.Thus we have n−2 2 + n − 1 = n−1 2 + 1 linearly independent vectors.We need only make sure that they generate C 0 .But this is clear since s 0 x =  + s 1 x + s 2 x , and Regarding the bounds on the minimum weight, for the upper bound we have wt(s 2 x ) = 3n − 8 as an upper bound, except when n = 8 and wt(s 0 x ) = 11 < 3n − 8 = 16.For the lower bound we use Lemma ?? which shows that the bound is at least n − 2 and equal to this only if w a,b ∈ C 0 .Thus suppose that w a,b ∈ C 0 .From the basis we have just obtained, w a,b = w + v where w ∈ C 2 and v ∈ w i | 1 ≤ i ≤ n − 1 .Now for all ∆ ⊂ Ω of size 4, (w a,b , w(∆)) = 0, where w(∆) is defined in Equation (??).Thus (v, w(∆)) = 0 for all ∆.Suppose v = i∈I w i , where I ⊆ Ω \ {n}.Notice that (w i , w(∆)) = 0 if i ∈ ∆, and 1 if i ∈ ∆.If |I| ≥ 3, suppose i, j, k ∈ I, ∆ = {n, i, j, k}; then (v, w(∆)) = 1, a contradiction.So 1 ≤ |I| ≤ 2, and since n ≥ 8, there exists j, k, l ∈ I, i ∈ I, so with ∆ = {i, j, k, l} we have (v, w(∆)) = 1, also a contradiction.Thus w a,b ∈ C 0 which establishes the lower bound.
To show that Hull(C 0 ) ⊂ E 0 , from Lemma ??, C 0 = E 0 + s 0 x for any x ∈ V .If w is in the hull but w ∈ E 0 then w = u + s 0 x where u ∈ E 0 .For any y ∈ V , (s 2 y , u) = 0 by Lemma ?? #2, so (w, Thus we have a contradiction, and deduce that Hull(C 0 ) ⊂ E 0 .
For the minimum weight of C ⊥ 0 , we know from Lemma ?? that this is at most 8.We have 2 by Corollary ??.The only weight-4 words in C ⊥ 2 have the form of Equation (??), and these are not in C ⊥ 0 .Thus C ⊥ 0 has minimum weight at least 6, and since it must be even, at most 8.But for n > 8, by Lemma ??, the only weight-6 words in C ⊥ 2 are those from Equation (??), and these are not in C ⊥ 0 .For n = 8 there are more weight-6 words but have the form w a,b and these are not in C ⊥ 0 .Thus C ⊥ 0 has minimum weight 8, since where z ranges over the basis for C 2 given in Proposition ??, excluding s 2 {1,2,3} , together with the words w i + w 1 for 2 ≤ i ≤ n − 1, and the single word s 0 {1,2,3} + .The words w a,b are not in C 0 .Further, C ⊥ 0 has minimum weight 8.
Proof: This can be proved by showing that a matrix similar to that in Proposition ?? can be put into row echelon form.In this case the rows correspond first to the n−2 2 − 1 words s 2 z + s 2 {1,2,3} for z from the vertices in the basis for C 2 as in Proposition ??, then the words w 1 + w i for 2 ≤ i ≤ n − 1, then finally s 0 {1,2,3} + .The columns are the same as those from C 1 of Proposition ??, except that we remove for now the column of {2, 3, n}.Next come the columns labelled C 4 before, followed by the column for {2, 3, n}.The columns C 2 and C 4 are then as before.
It can be shown that this reduces to a row echelon form such that each row has a leading entry and thus that the dimension of the space C spanned by these rows is the number of rows, i.e. n−1 2 .To show this is C 0 , notice first that the sum of the first n−2 2 − 1 rows is , so  ∈ C, and hence also s 0 {1,2,3} ∈ C. Since any word s 2 y + s 2 z can be obtained from the first n−2 2 −1 rows, they are all in C. Also we have w 1 +w n ∈ C, since  ∈ C, so all the sums s 1 y +s 1 z are in C. Since  = s 0 x +s 1 x +s 2 x , we have, for any x ∈ V , s 0 x +s 0 {1,2,3} = s 1 x +s 1 {1,2,3} +s 2 x +s 2 {1,2,3} ∈ C, and hence s 0 x ∈ C for all x, and C = C 0 .For the bounds on the minimum weight, as an upper bound we have wt(s 2 x + s 2 y ) = 4n − 16 when x 2 ∼ y.For the lower bound again we know from Lemma ?? that a word of weight n − 2 must be of the form w a,b .Thus suppose w a,b ∈ C 0 .Since C 0 is self-orthogonal, we must have (w a,b , s 0 x ) = 0 for all x ∈ V .Clearly this is not the case for x = {a, b, c}, so w a,b ∈ C 0 which establishes the lower bound, since C 0 is an even-weight code, so n − 1 is not possible.
Using Magma [?, ?], we found the following: Example 1.For n = 10, we know that the self-orthogonal code C 0 has dimension 36.From Magma it has minimum weight 24 and there are 5355 minimum words, in three orbits under Aut(Γ 0 10 ) ∼ = S 10 , and they can be described as follows: • 630 words of the form s 2 x + s 2 y where x 2 ∼ y; • 1575 words of the form • 3150 words of the form The supports of the 5355 minimum words form the blocks of a 2-(120, 24, 207) design D which is such that Aut(D) = Aut(C 0 ), and this group has order 47377612800 = 2 16 * 3 5 * 5 2 * 7 * 17, is simple, 2-transitive on points, primitive on blocks, and is isomorphic to the symplectic group Sp 8 (2) (or, alternatively, the simple orthogonal group SO 9 (2)).Thus the 5355 minimum words are in one orbit under Aut(C 0 ).The binary code C 2 (D) of the design has dimension 35, and does not contain the words s 0 x , although  ∈ C 2 (D), it being the sum of all the rows.Both C ⊥ 0 and C 2 (D) ⊥ have minimum weight 8, containing the words of weight 8 defined by partitions, as in Lemma ??.That the minimum weight of C 2 (D) ⊥ is at least 8 follows also from the design parameters, since the replication number r for D is 1071, so if a word in C 2 (D) ⊥ has weight s we must have s − 1 ≥ r/λ = 1071/207 = 5.2, so s ≥ 7 and since  ∈ C 2 (D), it is of even weight and thus s ≥ 8.The automorphism group of each of the other codes for n = 10, and of all the graphs and neighbourhood designs, is just S 10 .Example 2. For n = 7, Γ 0 7 is the odd graph O 3 , and Aut(C 0 ) = S 8 = Aut(N 1 7 ) = Aut(R 1 7 ) = Aut(Γ 1 7 ) = Aut(RΓ 1 n ) = Aut(C 1 ).The other groups are all S 7 .Extra automorphisms that then generate S 8 can easily be defined in this case, as was already described in [?].For a ∈ Ω = {1, . . ., 7}, let Ω a = Ω \ {a}, and for x ∈ V , let x ca = Ω a \ x.Then the map α a defined by is easily seen to be an automorphism of the graphs Γ 1 7 , RΓ 1 7 , and thus of their neighbourhood designs and codes.It was already shown to be an automorphism of the code C 0 in [?], since for n = 7, Γ 0 7 is an odd graph, O 3 .The α a are not automorphisms of the graph Γ 0 7 = O 3 .They are also not automorphisms of the Johnson graph Γ 2 7 .

Conclusion
The table shown is a summary of some of the facts we have established about the codes.The column labelled i refers to the codes from A i + I, for i = 0, 1, 2 and H i = Hull(C i ).The second column denotes the value of n modulo 4. The entries with * in the first and 9 th rows are for n > 8; for n = 8 the minimum weight of C A0+1 is 11 = |s 0 x | < |s 2 x | = 16, while the words of weight 16 are not only the rows s 2 x in C A2+1 .The entries with a † in the fourth row is for n > 7; for n = 7 there are words of weight n − 2 = 5 other than the w a,b , and the minimum weight of H 0 is 16: see Proposition ??, since H 0 = E 1 .An entry ?means we have not proved this, i.e. this is from Magma.For n ≡ 2 (mod 4), all the C i are self-orthogonal and thus equal to their own hulls.In the set of rows for i = 0, the entry 4(n − 4) is the weight of s 2 x + s 2 y when x 2 ∼ y, since in A 2 + I, rows meet in 0 points if |x ∩ y| = 0, 4 points if |x ∩ y| = 1, and in n points if x 2 ∼ y.The minimum weight for E 1 is for n ≥ 11; see Proposition ?? for 7 ≤ n ≤ 10.Note that the minimum weight of C i is at least n − 2 for all the i = 0, 1, 2, by Lemma ?? so we do not include this lower bound for the minimum weight in the table.
In the introduction we mentioned a series of codes W i , W Π over any F p that can be used to establish results about codes from the uniform subset graphs Γ(n, k, r) = (V, E).The W i are defined in the obvious way: if x ⊆ Ω and |x| = i, then the word w x = y∈V,x⊂y v y , and W i = w x | x ⊂ Ω, |x| = i , where the span is over F p .
For the code W Π we make use of partitions of subsets of size 2k of Ω.Let such a partition π be then the word w π will have the set of k-sets {a i1 , b i2 , . . ., k i k } as support with the sign being determined by giving x = {a 1 , b 1 , . . ., k 1 } the sign "+", and then demanding that any other k-set in the support with intersection of size k − 1 with x will have sign "−", and then applying this in general to get the signs on all the 2 k vertices.Alternatively the words can be defined inductively: for example, from the partition for k = 3 given in Definition ??, we can get to one for k = 4 with the extra partition set [d 1 , d 2 ] by adjoining d 1 to all the elements of the sets X and X c , keeping the same signs, and then do the same with d 2 , but switiching the signs.Another interpretation takes the 2 k vertices in the support of w π as the vertices of the k-cube, Q k , i.e. the Hamming graph H(k, 2), with alternate signs on the vertices.For all 1 ≤ i ≤ k − 1, W Π ⊆ W ⊥ i .These words were used in [?] in the binary case for codes from Johnson and odd graphs.The codes W i , W Π will come into play for all the codes, over any F p , from the adjacency matrix A i , A i + I, A i + I + J (for complementary graphs).

Lemma 9 .
For any n ≥ 7, any code C ⊆ F V 2 with W Π ⊆ C ⊥ , has minimum weight at least n − 2 and the words of weight n − 2 are the w a,b if n ≥ 8.This is true for C = W 2 .

Table 1 :
Binary codes of C i for n ≥ 7 Note: The design and code acted on by Sp 8 (2) is also constructed, in a different way, in [?].