Increasing the minimum distance of codes by twisting

Twisted permutation codes, introduced recently by the second and third authors, are frequency permutation arrays. They are similar to repetition permutation codes, in that they are obtained by a repetition construction applied to a smaller code. It was previously shown that the minimum distance of a twisted permutation code is at least the minimum distance of a corresponding repetition permutation code, but in some instances can be larger. We construct two new infinite families of twisted permutation codes with minimum distances strictly greater than those for the corresponding repetition permutation codes.


Introduction
Powerline communication has been proposed as a solution to "the last mile problem" in the delivery of reliable telecommunications at the lowest cost [11,15].A constant composition code of length m over an alphabet Q of size q has the property that each codeword has p i occurrences of the i th letter of the alphabet, where the p i are positive integers such that p i = m.Such codes have been proposed as suitable coding schemes to solve the narrow band and impulse noise problems associated with powerline communication [4,5].
One approach is to choose p i = 1 for all i , and in this case each codeword is a permutation on m letters.An (m, d) permutation array, usually denoted by P A(m, d), is a set of permutations on m letters with the property that the Hamming distance between any two distinct permutations in this set is at least d.A larger family of constant composition codes, which properly contains the permutation arrays, consists of those in which each letter occurs p times, where p = m/q .These codes are called frequency permutation arrays (FPA), and have been studied, for example, in [4,12,13].FPA's also play an important role in the study of neighbour transitive codes, introduced by the second and third authors [7].In particular they arise naturally in certain classifications of these codes [8,9].
Another practical application of FPA's includes the area of multilevel flash memories.Flash memory is an electronic non-volatile memory that uses floating-gate cells to store information [6].In flash memories, cells are organized into blocks, where each block has a large number (≈ 10 5 ) of cells [6].Given a set of n cells with distinct charge levels, the rank of a cell indicates the relative position of its own charge level, and so the ranks of the n cells induce a permutation of {1, 2, . . ., n} .Schwartz et al. [14,17] studied error-correcting codes for such permutations under the infinity norm, motivated by a novel storage scheme for flash memories called rank modulation (which uses these permutations).As for other applications of flash memory: Shieh and Tasi applied FPAs to provide multilevel flash memory with error correcting capabilities, and because of their efficient encoding and decoding algorithms, FPAs can be used in flash memory systems to represent information and correct errors caused by charge level fluctuation [16].
Twisted permutation codes are examples of frequency permutation arrays with potentially good minimum distance properties (see [10]).These codes are generated by groups and are generalisations of repeated permutation codes.In [10], the second and third authors, with Spiga, proved that the minimum distance of a twisted permutation code is at least the minimum distance of a corresponding repetition permutation code for the group.Moreover, they gave examples for which the minimum distance is greater than that achieved by the repetition construction (see [10,Table 1]).
In this paper, we give two new infinite families of twisted permutation codes with minimum distance strictly greater than the lower bound given by the usual repetition construction.We hope that the ideas behind these constructions might suggest further new constructions with equal or better improvements in minimum distance.We would be interested to know how much improvement in minimum distance is possible: can this be quantified?
Let T be an abstract group, let I be an ordered r -tuple of permutation representations of T on the set {1, . . ., q} (with repeats allowed), and let ρ be (any) one of these representations.In Section 2.3 we define the twisted permutation code C(T, I) and the repetition code Rep r (C(T, ρ)).Both are frequency permutation arrays of length rq over the alphabet {1, . . ., q} with each letter occurring r times in each codeword.Let δ tw denote the minimum distance of C(T, I), and let δ rep be the minimum of the minimum distance of Rep r (C(T, ρ)).We prove the following.
Theorem 1.1.The twisted permutation codes described in Table 1 have minimum distance δ tw strictly greater than δ rep .

2.1.
Codes in Hamming Graphs.For positive integers m, q , each at least 2 , the Hamming graph Γ = H(m, q) is the graph whose vertex set V (Γ) is the set of m-tuples with entries from an 'alphabet' Q of size q , such that two vertices form an edge if and only if they differ in precisely one entry.A code The automorphism group of Γ, which we denote by Aut(Γ), is the semi-direct product B ⋊ L where B ∼ = S m q and L ∼ = S m , see [3,Theorem 9.2.1].Its action is described as follows: for g = (g 1 , . . ., g m ) ∈ B , σ ∈ L and α = (α 1 , . . ., α m ) ∈ V (Γ), For all pairs of vertices α, β ∈ V (Γ), the Hamming distance between α and β , denoted by d(α, β), is defined to be the number of entries in which the two vertices differ.It is the distance between α and β in Γ, and so we usually refer simply to distance, rather than Hamming distance.The minimum distance, δ(C), of a code C is the smallest distance between distinct codewords of C , that is, If C consists of exactly one codeword, then we set δ(C) = 0 .A code C is called distance invariant if, for all positive integers i , the number of codewords at distance i from a codeword α ∈ C is independent of the choice of α .

Permutation Groups.
Let Ω be an arbitrary non-empty set.We denote by Sym(Ω) the group of all permutations of Ω, called the symmetric group on Ω.A permutation group on Ω is a subgroup of Sym(Ω).For t ∈ Sym(Ω) and α ∈ Ω, we denote by α t the image of α under t.Suppose that G is a permutation group on Ω and t ∈ G.We define the support of t by supp(t) = {α ∈ Ω : and the set of fixed points of t by fix(t) = {α ∈ Ω : α t = α}.
Then Ω = supp(t) ∪ fix(t) for all t ∈ G.The minimum value min{| supp(t)| : 1 = t ∈ G} is called the minimal degree of G.
Let G 1 and G 2 be two groups acting on the sets Ω 1 and Ω 2 , respectively.Then the two actions are said to be permutationally isomorphic if there exists a bijection λ : Ω 1 → Ω 2 and an isomorphism ϕ : The pair (λ, ϕ) is called a permutational isomorphism.
Then C(T, I) is a code in H(rq, q), and is called a twisted permutation code.In particular, if r = 1 then C(T, I) is the permutation code C(T, ρ 1 ) given in (2.2), and if Proposition 2.2.[10] With the notation as above, consider the code C = C(T, I).Then, The lower bound for δ(C(T, I)) given in Proposition 2.2 (iii) is the bound we wish to improve on!

The affine group
In this section we use affine groups to construct a family of twisted permutation codes with minimum distance greater than the lower bound of Proposition 2.2.Let k be an integer and p an odd prime number such that p > k 2 , and let V = F k p be the vector space of k -dimensional row vectors over the finite field F p .Let A k denote the k × k matrix over F p whose i th row is equal to where e i denotes the i th standard basis vector of V .In particular we have that where T denotes matrix transpose.Now define where I k is the k × k identity matrix over F p .
Lemma 3.1.For 2 k < p and any positive integer i , we have Proof.First note that by multiplying together the two expressions for A k in (3.1) we find that Then observe that, by induction, for 1 s k − 1 , Thus A k k = 0, the zero matrix.Now for any positive integer i we evaluate Using the expression for A s k above and the fact that A k k = 0, we deduce the expression for B i k .Since k < p, it follows that B p k = I k .
For a vector v = (v 1 , v 2 , . . ., v k ) ∈ V , we denote by ϕ v the translation by v , and for each k 2 , we define We also set Ω(k, 1) := I k and, for a positive integer i 2 , we set Ω(k, i) With the notation above, we have for any positive integer i and 2 k < p, where for 1 j k , i j inside the matrix denotes 1 + . . .+ 1 ( i j ) in F p .In particular, Ω(k, p) = 0.
Proof.First we note the following binomial identity (see, for example, [2, Identity 135]): for 0 ℓ i−1 , Using this, and Lemma 3.1, it follows that a s,t , the s, t entry of Ω(k, i), is equal to giving us the required expression.Since k < p, we deduce that Ω(k, p) = 0, the zero matrix.
Remark 3.3.We observe that for any positive integers i, j , Ω(k, i + j) = Ω(k, i) + B i k Ω(k, j).Since B p k = I k and Ω(k, p) = 0, this implies that for all i , Ω(k, i) = Ω(k, i mod p), where i mod p denotes the integer i 0 in {1, . . .p} such that i − i 0 is divisible by p.
To each w ∈ V we associate a map τ w : G k → G k which, for each v ∈ V and each i 1 , takes Proof.First of all, we show that τ w is a homomorphism: for each v 1 , v 2 ∈ V and for each i, j , and hence B i k = I , which implies by Lemma 3.1, Lemma 3.2 and Remark 3.3 that i = p mod p, Ω(k, p) = 0 and v = 0. Thus ϕ v B i k = I .Finally, we show that τ w is onto: for an arbitrary element Then the map λ : V → Ω given by v → (1, v) is a bijection, and we deduce the following.
Proposition 3.5.The pair (λ, τ 0 ) is a permutational isomorphism from the action of G k on V to the action of G K on Ω.
Proof.From the definition of λ we have On the other hand, it follows from the definition of τ 0 that , and hence (λ, τ 0 ) is a permutational isomorphism.
We will need the following information about fixed points of elements in this action.
3), and let with vB i k = (v 1 , v 2 , . . ., v k ) ∈ V for some i 1 .Then g fixes exactly 0 or p points in Ω.Moreover, (1) g fixes p points ⇐⇒ i = p mod p and v k = 0 ; and (2) g has no fixed points ⇐⇒ i = p mod p, or v k = 0 .
Proof.Suppose g has a fixed point (1, x) where x = (x 1 , x 2 , . . ., x k ).Then by Lemma 3.1 and some easy calculations, we obtain Comparing the entries in both sides of the equation, we see that (3.6) x j = v j + k−j t=0 i t x t+j .for j = 1, . . ., k .
In particular, (3.6) with j = k implies that v k = 0 .We claim that i = p mod p. Suppose to the contrary that i = p mod p. Then (3.6) implies that x j = v j + x j for all j , so v = 0.In particular, g = 1 , which is a contradiction.Hence, if g fixes a point then i = p mod p and v k = 0 .
Conversely suppose that i = p mod p and v k = 0 .A point (1, x) is fixed by g if and only if the equations (3.6) hold.Now i mod p is invertible in F p (which we identify with i here), so (3.6) imply that (3.7) Thus the entries x 2 , . . ., x k are determined by vB i k = (v 1 , v 2 , . . ., v k ), while x 1 is unrestricted; so g fixes exactly p points of Ω, namely (1, a, x 2 , . . ., x k ) for a ∈ F p .This completes the proof.
Let w r := (0, 0, . . ., 0, r) ∈ V where r ∈ F p .Note that, if r = 0 , then the composition of the isomorphisms τ wr and τ −1 w0 , namely τ wr τ −1 w0 , is an automorphism of G k .We need the following information about these automorphisms.(i) If i = p mod p then for each r ∈ F p , τ wr τ −1 w0 (g) has no fixed points in Ω. (ii) If i = p mod p then there exists a unique r ∈ F p \ {0} such that τ wr τ −1 w0 (g) fixes p points of Ω, and for each s = r , τ ws τ −1 w0 (g) has no fixed points in Ω.
Proof.It follows from Lemma 3.4 that τ wr τ −1 w0 is an automorphism of G k .Direct calculation gives If i = p mod p, then τ wr τ −1 w0 (g) = g , and hence, by Lemma 3.6, τ wr τ −1 w0 (g) has no fixed points in Ω for all r ∈ F p , proving part (i).Now suppose i = p mod p. Then where uB i k = (u 1 , u 2 , . . ., u k ) = w r Ω(k, i) + vB i k .Thus, using Lemma 3.2, we obtain By Lemma 3.6, τ wr τ −1 w0 (g) fixes p points if and only if u k = v k + ir = 0 .Since, i = p mod p we can identify i with i mod p ∈ F p \{0} , and so the previous expression holds if and only i , then τ ws τ −1 w0 (g) has no fixed points in Ω, proving part (ii).
Table 2 summarises the information about the support sizes of τ wr τ −1 w0 (g) derived in the proof of Lemma 3.7.
We define the permutation code and note that C(T ) is equal to {α(τ (g))|g ∈ T } for each τ ∈ Aut(T ).We define the twisted permutation code relative to I as C(T, I) = {α(g, I)|g ∈ T }.Proposition 3.8.Let T = G k , I , and C(T, I) be as above.Then the minimum distance of C(T, I) is p k+1 − p, while the lower bound given by Proposition 2.2 for this code is p k+1 − p 2 .Proof.By Lemma 2.1 and Table 2, the minimum distance of C(T ) is p k − p for each r .Hence the lower bound on the minimum distance of C(T, I) given by Proposition 2.2(iii), namely the minimum distance of the p-fold repetition code generated by C(T ), is p k+1 − p 2 .On the other hand, by Lemma 3.7 and Table 2, for each g ∈ T as in (3.5), Thus by Proposition 2.2, δ(C(T, I)) = p k+1 − p which is strictly greater than the lower bound in Proposition 2.2 (iii).

The symplectic group
In this section, we consider the symplectic group T = Sp(4, q) over a field of order q = 2 n 2 .We exploit the fact that G has an outer automorphism τ which does not map transvections to transvections.We preserve this notation throughout.Set I = {ι, τ } where ι denotes the identity map.We show that the twisted permutation code C(T, I) has minimum distance strictly greater than the lower bound in Proposition 2.2.
Let V = F 4 , the space of 4 -dimensional row vectors over a field F of order q = 2 n , and write the set of 1 -dimensional subspaces of V .For g ∈ GL(4, q), we denote by the subset of 1 -spaces fixed setwise by g .Now V admits a non-degenerate symplectic form with automorphism group T = Sp (4, q).Let e 1 , e 2 , f 1 , f 2 be a symplectic basis for V such that {e 1 , f 1 } and {e 2 , f 2 } are hyperbolic pairs, so that Then M := e 1 , e 2 is a maximal totally isotropic subspace with dimM = 2 = (dimV )/2 .We begin by stating without proof the following result concerning the isometry groups of nondegenerate symplectic bilinear forms.Versions of this theorem go back to Witt's work in [20] and all versions are commonly referred to as Witt's Theorem; the proof of the below statement may be found in the book of Artin [1, p.121].
Theorem 4.1.Let U be a subspace of a non-degenerate symplectic space V , and let f : U → V be a linear isometry.Then f can be extended to a linear isometry of V , that is, there is a linear isometry h : V → V such that f (u) = h(u) for all u ∈ U .Lemma 4.2.Let g ∈ T = Sp(4, q) and suppose that w | w ∈ Fix P G(V ) (g) has dimension at least 3 .Then g is conjugate by an element of T to an element whose matrix, with respect to the ordered basis (e 1 , f 1 , e 2 , f 2 ), has the following form Proof.By assumption V contains three linearly independent vectors v 1 , v 2 , v 3 such that g fixes each v i setwise, and thus g fixes setwise W := v 1 , v 2 , v 3 .Since W is of odd dimension, its radical R := W ∩W ⊥ (where W ⊥ = {w ∈ W | (w, u) = 0 for all u ∈ W } ) must be non-zero, and also R < W , because the maximum dimension of the totally isotropic subspaces of V is equal to 2. Thus W/R is nontrivial and non-degenerate, and hence dim(W/R) = 2 and dim R = 1 .Further, since R ⊆ W ⊥ , and since V is non-degenerate, we have W = R ⊥ .Now there is a linear isometry R → V which maps R to e 1 , and by Theorem 4.1 this extends to an element of T mapping R to e 1 .Replacing g , if necessary, by its conjugate under this element, we may assume that R = e 1 and hence that W = R ⊥ = e 1 , e 2 , f 2 .
We claim that the setwise stabiliser Stab T (W ) is transitive on the 1 -spaces contained in W \ R .To see this, let w 1 and w 2 be linearly independent vectors in W \ R .Then U i := e 1 , w i is totally isotropic for i = 1, 2 .There is a unique linear map f : and this map f is a linear isometry, because U i is totally isotropic for i = 1, 2 .Therefore Witt's Theorem implies that there exists y in T which sends Now y fixes e 1 , and hence also e 1 ⊥ = W , so y ∈ Stab T (W ) and the claim is established.
In what follows, we may assume without loss that v 1 ∈ W \ R .By the previous paragraph, there exists an element x ∈ Stab T (W ) such that x sends v 1 → e 2 and e 1 → e 1 .So g x = x −1 gx fixes e 2 setwise.Hence g x fixes e 1 , e 2 , and W . Since g x is a conjugate of our original element g , there exists a vector w ∈ W \ e 1 , e 2 such that w is fixed by g x .Replacing w by a scalar multiple if necessary, we may assume that w = ce 1 + de 2 + f 2 , for some c, d ∈ F. Now W = e 1 , e 2 , f 2 = e 1 , e 2 , w and it is straightforward to check that the map φ : W → V which sends φ : e 1 → e 1 , e 2 → e 2 , f 2 → w, defines a linear isometry.Hence by Witt's Theorem, there exists an element say y ∈ T such that y : e 1 → e 1 , e 2 → e 2 , f 2 → w.
Then g xy −1 = yg x y −1 fixes e 1 , e 2 , and f 2 .Now we replace g by its conjugate g xy −1 .Then g fixes W , e 1 , e 2 , and f 2 , and so, for some a, b, c ′ and d i , Since g preserves the form and (e 2 , f 2 ) = 1 , (e 1 , f 1 ) = 1 , (e 2 , f 1 ) = 0 , and (f 1 , f 2 ) = 0 , we obtain c ′ = b −1 , d 2 = a −1 , d 4 = 0 , and d 3 = 0 .This shows that the matrix for g with respect to the ordered basis (e 1 , f 1 , e 2 , f 2 ) is as in (4.1).Lemma 4.3.Suppose that q is even and that g ∈ Sp(4, q) with g = 1 .If g is a transvection, then g has exactly q 2 + q + 1 fixed 1 -spaces in V ; otherwise it has at most 2q + 2 fixed 1 -spaces in V .Proof.It is not difficult to see that if a non-scalar element g ∈ GL(4, q) fixes v 1 , v 2 setwise and g fixes v 1 and v 2 , then the number of 1 -spaces in v 1 , v 2 fixed setwise by g is 2 or q + 1 .Thus if all the 1 -spaces fixed setwise by g lie in a 2 -space, then there is nothing to prove.So we assume that g fixes setwise three 1 -spaces v 1 , v 2 , and v 3 such that v 1 , v 2 , v 3 are linearly independent.Then g fixes W := v 1 , v 2 , v 3 setwise, so, by Lemma 4.2, conjugating g by an element of Sp(4, q) if necessary, we may assume that the matrix for g with respect to the ordered basis (e 1 , f 1 , e 2 , f 2 ) is as in (4.1).
, and suppose that g fixes v setwise, so vg = tv for some t ∈ F \ {0} .Then We find Fix PG(V ) (g) and its size according to the possibilities for |S| and d.Assume first that d = 0 .In this case |S| 2 since g = 1 and the only scalar matrix in Sp(4, q) is the identity.
As pointed out by Todd [19], using Lemma 4.4 one can obtain some isomorphisms between Sp(4, 2 n ) and O(5, 2 n ).However the geometric reasons for these isomorphisms are quite different.Taylor [18, p. 201] observes that the exterior square Λ 2 V of V = F 4 has dimension 4 and admits a nondegenerate alternating form which is invariant under Sp(4, q).Moreover each element g of Sp(4, q) induces a linear map of Λ 2 V , which we denote by Λ 2 g , giving a second representation of Sp(4, q) in dimension 4 .
Let t be a transvection in Sp(4, q).Then Taylor shows [18, p. 202] that the map 1 − Λ 2 t of Λ 2 V has image of dimension 2 , and hence also is a 2 -dimensional subspace of Λ 2 V .Since Λ 2 t is unipotent, each 1 -space fixed setwise by Λ 2 t must be fixed pointwise, and hence must lie in Fix Λ 2 V (Λ 2 t).Thus Λ 2 t fixes setwise exactly q + 1 of the 1-spaces in Λ 2 V , and hence it does not act as a transvection on Λ 2 V .
Taylor composes the representation Λ 2 with a linear map Λ 2 V → V to obtain an outer automorphism τ of Sp(4, q) under which t is mapped to an element tτ of Sp(4, q) with exactly q + 1 fixed 1-spaces in V .Thus we have the following result.Lemma 4.5.There exists an outer automorphism τ of Sp(4, q) under which the image of a transvection is an element with q + 1 fixed 1 -spaces.This is the information we need to analyse the twisted permutation code C(T, I) for T = Sp(4, q) relative to I = (ι, τ ), with ι the identity automorphism of T and τ the outer automorphism from Lemma 4.5 .To construct this code, we let V = F 4 , where F is a field of order q , we set m := (q 4 − 1)/(q − 1), and we choose an ordering ( v 1 , v 2 , . . ., v m ) for the set PG(V ) of 1 -dimensional subspaces in V .Each g ∈ T then corresponds to the vertex α(g) = ( v 1 g , v 2 g , . . ., v m g ) of the Hamming graph of length m over PG(V ).If we identify T with the permutation group it induces on PG(V ) then ι and τ can be interpreted as permutation representations of T on PG(V ).We define the permutation code as Proposition 4.6.Let T = Sp(4, q) and I = (ι, τ ) be as above.Then the twisted permutation code C(T, I) has minimum distance 2q 3 + q 2 , and the difference between this minimum distance and the lower bound given by Proposition 2.2 is q 2 .Proof.By Proposition 2. which is strictly greater than 2q 3 + q 2 for q 4 .Therefore, δ(C(T, I)) = 2q 3 + q 2 .Finally, we have from Proposition 2.2 and Lemma 4.3 that the lower bound in Proposition 2.2 is the minimum of 2| supp(g)| for nontrivial g ∈ T , and that this is attained when g is a transvection and is 2(m − q 2 − q − 1) = 2q 3 .The difference between δ(T, I) and this lower bound is q 2 .