ON BLOCKING SETS IN PROJECTIVE HJELMSLEV PLANES

. A ( k,n )-blocking multiset in the projective Hjelmslev plane PHG( R 3 R ) is deﬁned as a multiset K with K ( P ) = k , K ( l ) ≥ n for any line l and K ( l 0 ) = n for at least one line l 0 . In this paper, we investigate blocking sets in projective Hjelmslev planes over chain rings R with | R | = q m , R/ rad R ∼ = F q , q = p r , p prime. We prove that for a ( k,n )-blocking multiset with 1 ≤ n ≤ q , k ≥ nq m − 1 ( q +1). The image of a ( nq m − 1 ( q +1) ,n )-blocking multiset with n < char R under the the canonical map π (1) is “sum of lines”. In particular, the smallest ( k, 1)-blocking set is the characteristic function of a line and its cardinality is k = q m − 1 ( q + 1). We prove that if R has a subring S with p | R | elements that is a chain ring such that R is free over S then the subplane PHG( S 3 S ) is an irreducible 1-blocking set in PHG( R 3 R ). Corollaries are derived for chain rings with | R | = q 2 , R/ rad R ∼ = F q . In case of chain rings R with | R | = q 2 , R/ rad R ∼ = F q and n = 1, we prove that the size of the second smallest irreducible ( k, 1)-blocking set is q 2 + q + 1. We classify all blocking sets with this cardinality. It turns out that if char R = p there exist (up to isomorphism) two such sets; if char R = p 2 the irreducible ( q 2 + q + 1 , 1)-blocking set is unique. We introduce a class of irreducible ( q 2 + q + s, 1) blocking sets for every s ∈ { 1 ,...,q + 1 } . Finally, we discuss brieﬂy the codes over F q obtained from certain blocking sets.


Introduction
The motivation for this work comes from two sources -coding theory and finite geometry. In the past ten years, a substantial research has been done on linear codes over finite rings. It has been fuelled by the discovery that certain nonlinear codes that perform better than any linear codes over a finite field are in fact images of linear codes over the ring Z 4 [8,24]. Attempts have been made to obtain a theory of error-correcting codes over a reasonable class of rings (cf. [7,21,25,28]). In [11] and [13], the theory of linear codes over the class of so-called finite chain rings was developed. Various results from [8,11,24,26,27] show that using "good" linear codes over chain rings one can construct "good" (not necessarily linear) codes over finite fields.
It turns out that many good properties known for linear codes over finite fields still hold for linear codes over finite chain rings. In particular, there is an one-to-one correspondence between the classes of equivalent multisets of points in the projective Hjelmslev geometries and the classes of semilinearly isomorphic fat linear codes over the chain ring R [13]. With each multiset in PHG(R k R ), we can associate a linear code over R by taking a matrix with columns the points of this multiset written in homogeneous coordinates as a generator matrix. Conversely, each fat linear code over a finite chain ring R gives a multiset of points in PHG(R k R ). Moreover, two multisets are equivalent if and only if the corresponding codes are semilinearly isomorphic. The task of constructing multisets of maximal cardinality in projective Hjelmslev geometries containing no more than a prescribed number of points in each hyperplane very often leads to the construction of linear codes over chain rings with interesting properties.
The problem of constructing optimal multisets of points in certain finite geometries is interesting in its own right and is older than its coding theoretic counterpart. There is a vast literature about sets of points in the classical projective geometries over finite fields (cf. [9,10] and the references there), but there are almost no results on such sets in finite Hjelmslev geometries. In this paper, we investigate blocking multisets in projective Hjelmslev planes PHG(R 3 R ), where R is a chain ring with |R| = q m and R/ rad R ∼ = F q . We confine ourselves to right Hjelmslev planes PHG(R 3 R ). This is no restriction since every left R-module is a right module over the opposite ring R opp , which is also a chain ring with q m elements and residue field of order q.
In section 2, we introduce projective Hjelmslev planes and present some results on the structure of planes obtained from finite chain rings. In section 3, we define arcs and blocking multisets and present a general lower bound on the minimal size of a blocking multiset with given parameters. It is a reformulation of a upper bound known for arcs over chain rings R with |R| = q m , R/ rad R ∼ = F q . In Section 4, we prove that for every (k, n)-blocking multiset in PHG(R 3 R ), n = 1, . . . , q, |R| = q m , R/ rad R ∼ = F q , k ≥ nq m−1 (q + 1). In case of equality, and n < char F q the image of the blocking set under the canonical map π (1) is a "sum of lines" (Theorem 2). Further, we prove that if R has a subring S with |R| elements that is a chain ring and such that R is free over S then the subplane PHG(S 3 S ) is an irreducible 1blocking set (Theorem 3). Two corollaries are derived for chain rings with |R| = q 2 , R/ rad R ∼ = F q . In Section 5, we prove that the size of the second smallest irreducible (k, 1)-blocking set in PHG(R 3 R ), where |R| = q 2 , R/ rad R ∼ = F q , is q 2 + q + 1. Moreover, in projective Hjelmslev planes over chain rings of characteristic p there exist (up to isomorphism) two such blocking sets, while in planes over chain rings of characteristic p 2 there is only one irreducible (q 2 +q+1, 1)-blocking set (Theorem 4). In section 6, we compute the parameters and the weights of various codes obtained from the linear codes associated with the blocking sets from the previous sections.
An important class of projective Hjelmslev planes is obtained from the so-called finite chain rings (cf. [3,22,23]). Let R be finite chain ring. Denote by P and L, respectively, the sets of all free rank 1 and all free rank 2 submodules of R 3 R . Define I ⊆ P × L by set-theoretical inclusion. It is easily checked that (P, L, I) satisfies (PH1)-(PH3) and is a projective Hjelmslev plane. We denote this plane by PHG(R 3 R ). Since every chain ring R is a quasi-Frobenius ring [5], the usual duality properties known from finite-dimensional vector spaces over fields hold also for R-modules. Hence every line l ∈ L can be considered as the set of free rank 1 submodules R satisfying the linear equation where at least one of the r i 's is a unit. In other words, the dual Hjelmslev plane (L, P, I ′ ) (where I ′ is the transpose of I) is isomorphic to PHG( R R 3 ). Let R be a chain ring with |R| = q m , R/ rad R ∼ = F q . We consider the projective Hjelmslev plane Π = (P, L, I) = PHG(R 3 R ). Two points X = xR and Y = yR are called i-neighbours, i = 0, 1, . . . , m, if |X ∩ Y | ≥ q i . This fact is denoted by X ⌢ ⌣ i Y . Two lines s and t are i-neighbours if for every X on s there exists a point Y on t with X ⌢ ⌣ i Y , and conversely, for every Y on t there exists X on s with Y ⌢ ⌣ i X. Every two points (lines) are 0-neighbours; 1-neighbourhood is the same as the neighbour relation defined by (N1) and (N2).
For every i ∈ {0, 1, . . . , m}, the relation ⌢ ⌣ i is an equivalence relation on P, as well as on L. The equivalence classes of this relation are denoted by [X] (i) , X ∈ P, respectively [s] (i) , s ∈ L. The set of all equivalence classes of ⌢ ⌣ i on points, resp. lines, is denoted by P (i) , resp. L (i) . Denote by π (i) the natural homomorphism Below we state some facts on the combinatorics and the structure of the projective Hjelmslev planes PHG(R 3 R ) (cf. [1,4,6,12,17,18,19,20,30]).  (e) if X ⌢ ⌣ i Y and X ⌢ ⌣ i+1 Y , X, Y ∈ P, i = 0, 1, . . . , m, there exist exactly q i lines incident with both X and Y ; by duality, if s ⌢ ⌣ i t and s ⌢ ⌣ i+1 t, s, t ∈ L, there exist exactly q i points incident with both s and t; (f ) the number of i-neighbour classes of points (lines) is q 2(i−1) (q 2 + q + 1).
Given a projective Hjelmslev plane Π = PHG(R 3 R ), we define the affine Hjelmslev plane AHG(R 2 R ) as an incidence structure with points -all points not incident with a fixed 1-neighbourhood class [l] of lines, with lines -all lines not in [l], and with incidence -the one inherited from Π. Equvalently, we define the points as all pairs (a, b), a, b ∈ R, the lines as all cosets of free rank 1 submodules of R 2 R , and incidence by set-theoretical inclusion. The lines of AHG(R 2 R ) can be partitioned into q m−1 (q + 1) classes of parallel lines.
Let us fix a point P ∈ P and denote by L (i) (P ) the set of all lines of L that are incident with at least one point from [P ] (i) . For two lines s, t ∈ L (i) (P ), we write s ∼ t if they coincide on [P ] (i) . Denote by L ′ a set of lines from L (i) (P ) that contains exactly one representative from each equivalence class under ∼. For any X ∈ P and any l ∈ L, we write X ⌢ ⌣ i l if there exists a point Y ∈ P such that (Y, l) ∈ I and X ⌢ ⌣ i Y . It is clear that in any projective Hjelmslev plane Let l be a line in Π and let i ∈ {0, 1, . . . , m} be fixed. We define a set of points P (i) by Further define an incidence relation J ⊆ P (i) × L by For two lines s, t ∈ L, we write s ∼ t if they are incident under J with the same elements of P (i) . We denote by L (i) a set of lines containing exactly one representative from each equivalence class of L under ∼. Set J (i) = J| P (i) ×L (i) .

Multisets of Points in Projective Hjelmslev Planes
Let Π = (P, L, I) be a projective Hjelmslev plane. Any mapping from the pointset P to the nonnegative integers K : P → N 0 is called a multiset in Π. The integer K(P ), P ∈ P, is called the multiplicity of P . The mapping K induces a mapping on the subsets of P by Here, the induced mapping is denoted (by a slight abuse of notation) again by K. The integer |K| = K(P) is called the cardinality or the size of K. A set of points Q with K(Q) = i is called an i-set with respect to K. In particular, points of multiplicity i are i-points and lines of multiplicity i are i-lines. The support supp K of a multiset K is the set of points of positive multiplicity: supp K = {P ∈ P | K(P ) > 0}.
Two multisets K ′ and K ′′ in the projective Hjelmslev plane Π are said to be equivalent if there exists a collineation σ in Π such that K ′ (P ) = K ′′ (σ(P )) for every point P ∈ P.
A (k, n)-blocking multiset K is called reducible if there exists (k ′ , n)-blocking multiset K ′ with k ′ < k and K ′ (P ) ≤ K(P ) for every point P ∈ P. A blocking multiset that is not reducible is called minimal or irreducible.
An arc (resp. a blocking multiset) K with K(P ) ∈ {0, 1} for every P ∈ P is called a projective arc (respecively, a projective blocking multiset, or simply a projective blocking set ). Projective arcs and projective blocking sets can be considered as sets of points by identifying them with their support.
Here i j denotes the number of points X for which there exists a point Y on l with X ⌢ ⌣ j Y , but for which there is no point Z on l with X ⌢ ⌣ j+1 Z. The sequence a = (a (i1,i2,...,im) ), (i 1 , i 2 , . . . , i m ) ∈ N m , where a (i1,i2,...,im) is the number of lines of type (i 1 , i 2 , . . . , i m ), is called the spectrum of K.
For any n ∈ N, we denote by κ n (R 3 R ) the minimal value of k for which there exists a (k, n)-blocking multiset in PHG(R 3 R ). For chain rings with |R| = q 2 , R/ rad R ∼ = F q , we can reformulate the upper bound for arcs from [14,15] to get . Thus results about arcs can be translated into results about blocking sets an vice versa. Traditionally, we consider blocking sets if n is "big", i.e. if n is "close" to the line size q(q + 1).

General Results on Blocking Sets in Projective Hjelmslev Planes
Until the end of this section, R will be a chain ring with |R| = q m and R/ rad R ∼ = F q . As before rad R = Rθ for some θ in rad R \ rad 2 R. The following theorem is based on the nested structure of projective Hjelmslev planes and provides a large class of blocking sets. Theorem 1. Let R be a chain ring. Let there exist blocking sets with param- Then there exists a (k 1 k 2 , n 1 n 2 )-blocking set in P HG(R 3 R ). Such blocking sets are not minimal in general. Henceforth, they are considered as trivial.
It turns out that for n ≤ q we can find the minimal size of a projective (k, n)blocking set and provide a characterization for the blocking sets with n < p = charF q . This result is known for projective Hjelmslev planes over chain rings R with |R| = q 2 , R/ rad R ∼ = F q (see e.g. [15]). Below we prove a generalization for arbitrary chain rings. We start with a lemma.
Then for every neighbour class on points [P ] = [P ] (1) with K([P ]) = a < q m−1 and every neighbour class on Proof. We use induction on m. The case m = 1 is trivial. Assume that m = 2. The structure induced on [P ] is isomorphic to AG(2, q) (cf. Now assume that our assertion has been proved for every s ≤ m − 1 and every (k ′ , n ′ )-blocking multiset with n ′ ≤ q s−1 in a plane PHG(S 3 S ), where |S| = q s , S/ rad S ∼ = F q . Further assume that R is a chain ring with q m elements and residue field of order q. Set Π = PHG(R 3 R ) and let K be a (k, n)-blocking multiset in Π with n ≤ q m−1 . Consider the incidence structure having as points all 2-neighbour classes contained in [P ] (1) , as lines the 2-neighbour classes of lines containing points from [P ] (1) and the incidence inherited from (P (2) , L (2) , J (2) ). This structure is isomorphic to AG(2, q) by Facts 2 and 3. The class of parallel lines in this affine geometry that have the direction of l is of cardinality q and, as above, at least one of these lines has multiplicity less than q m−2 . Let this line be incident with the points (1) . There If a < q m−1 we count the multiplicities of all line classes [l] through [P ] in Π (1) . By Lemma 1, again a contradiction. Lemma 3. Let K be a multiset in AG(2, q) such that K(l) = n < p = char F q for the lines of all but one parallel class. Then K is the sum of n (not necessarily different) lines of the missing class.
Proof. Extend AG(2, q) to PG(2, q) and let P 0 be the point on the infinite line that corresponds to the missing parallel class. Define K by Clearly K is a (n(q + 1), n)-blocking multiset in PG(2, q) and the result follows by Theorem 5 from [15]. We prove the lemma by induction on m and j. For m = 2, j = 1, the statement is contained in Theorem 6 from [15]. For j = 0 and any m the statement is equivalent to the fact that that any line in [l] has the same multiplicity n. Now assume that the statement is true for any (nq s−1 (q + 1), n)-blocking set in any projective Hjelmslev plane over a chain ring S with |S| = q s , s ≤ m − 1 S/ rad S ∼ = F q for all j, as well as for such a blocking set in Π = (P, L, I) = PHG(R 3 R ) where R is a chain ring with |R| = q m , R/ rad R ∼ = F q , and every 0 ≤ j < j 0 ≤ m − 1. We have to prove that Consider an 1-neighbour class of points (with respect to the induced neighbourhood in Π (j0) . It contains the points of q j 0 -neighbour point classes (with respect to the neighbourod in Π), [Q 1 ] (j0) , . . . , [Q q ] (j0) say. Now the (nonempty line segments t ∩ [Q i ] (j0) form an affine plane isomorphic to AG(2, q). The restriction of K to this affine plane has the property that all lines in all but one direction are blocked exactly n ′ times. By Lemma 3, this restriction is the sum of n ′ parallel lines having the exceptional direction. The lines in this parallel class are the sets of 'points' −1) ). The sum on the right contains q constant terms and is, therefore, divisible by q. To complete the proof, we use once again induction on m. Consider the geometry (P Theorem 2. Let R be a chain ring with |R| = q m , R/ rad R ∼ = F q and let K be a (k, n)-blocking multiset with 1 ≤ n ≤ q, in Π = PHG(R 3 R ). Then k ≥ nq m−1 (q + 1).

For every 1-neighbour class of lines [ℓ]
since every point is incident with q m−1 lines from [ℓ] (1) . There are exactly q 2(m−1) lines in [ℓ] (1) and thus (3) implies H([ℓ] (1) ) ≥ n. This gives in turn that H is an (n(q + 1), n)-blocking set in the projective plane (P (1) , L (1) , J (1) ) ∼ = PG(2, q). We complete the proof by applying once again Theorem 5 from [15]. Remark 1. It is impossible to replace the statement in this theorem that K (1) /q m−1 is the sum of lines by the stronger "K (i) /q m−i is the sum of lines for some i > 1". For instance, consider the following example in Π = PHG((Z 8 ) 3 Z8 ) which is easily generalized to the geometry over chain rings of arbitrarily high index of nilpotency. Let K be the (24, 2)-blocking set containing the following points (x 1 , x 2 , x 3 ): -all points from the line x 1 = 0; -all points on the line x 2 = 0 that are not in the point class [P ] (1) , where P = (0, 0, 1); -all points on the line X 1 + 2x 2 + 2x 3 = 0 that are in the point clas [P ] (1) .
It is a straightforward check that 1 2 K (2) is not the sum of two lines in Π (2) . Corollary 1. Let K be a (q m−1 (q + 1), 1)-blocking set in Π = PHG(R 3 R ), where R is a chain ring with |R| = q m , R/ rad R ∼ = F q . Then supp K is a line in Π.
Proof. As in Lemma 4, the multiset K = 1 q K (m−1) is a (q m−2 (q + 1), 1)-blocking set and hence its support is a line in Π (m−1) , [l] (m−1) say. The nonempty sets supp K ∩ [P ] (m−1) are line segments having the same direction as l. Finally, considering the factor geometry (P (1) , L (1) , J (1) ) defined for l (Fact 4), we prove by induction on m that these line segments are collinear with the same line in Π.
Theorem 3. Let R be a chain ring with |R| = q m , R/ rad R ∼ = F q , where q m is a perfect square. Let there exist a subring S of R that is a chain ring with |S| = √ q m and such that R is free over S. Then the multiset K defined by . Proof. Define ξ by R = S ⊕ ξS. This is possible by the fact that R is free over S. Every line in PHG(R 3 R ) can be considered as the set of points (x 1 , x 2 , x 3 )R that are solutions to: where a, b, c ∈ R and at least one of them is a unit. We have to show that there exists a solution to (4) with x 1 , x 2 , x 3 ∈ S such that at least one of the x i 's is a unit. Without loss of generality, we set a = 1, b = s 1 + ξt 1 , c = s 2 + ξt 2 . By (4), we get (x 1 + s 1 x 2 + s 2 x 3 ) + ξ(t 1 x 2 + t 2 x 3 ) = 0, which is equivalent to If we assume that Rt 1 = Rθ i , Rt 2 = Rθ j with 0 ≤ i ≤ j, then the submodule of R R 3 generated by (1, s 1 , s 2 ) and (0, t 1 , t 2 ) is of shape (m, m − i, 0). In other words, the submodule in question is isomorphic to R/N m ⊕R/N m−i ⊕R/N 0 , where N = rad R and N 0 = R The dual submodule is of shape (m, i, 0) and, therefore, has a free rank 1 submodule (cf. [13]). This implies the existence of a solution with the required property. Another possibility is to set x 3 = 1 (note that j ≥ i) and compute x 1 and x 2 from the linear system above.
Corollary 2. Let R be a chain ring with |R| = q 2 , R/ rad R ∼ = F q , that contains a subring S isomorphic to the residue field F q . Then PHG(R 3 R ) contains a subplane Γ isomorphic to PG(2, q) and the projective multiset K defined by supp K = Γ is an irreducible (q 2 + q + 1, 1)-blocking set.
The blocking sets described in Corollary 2 were introduced in the paper [2] in a slightly different context. They are defined as the orbit of a fixed point with coordinates from the residue field under a Singer cycle of PG (2, q). It turns out that the linear codes associated with these multisets can be mapped (cf. [11]) to two weight linear codes over F q . These in turn give rise to a family of strongly regular graphs with parameters v = q 6 , k = q 4 − q, λ = q 3 + q 2 − 3q, µ = q 2 − q.
The geometric structure of these blocking sets can be easily deduced from their definition. Every point X from supp K is incident with q + 1 lines of multiplicity q + 1 and on q(q + 1) − (q + 1) = q 2 − 1 lines of multiplicity 1. The number of all lines blocked by points from supp K is (q 2 + q + 1)(q + 1) q + 1 + (q 2 + q + 1)(q 2 − 1) = q 2 (q 2 + q + 1), i.e. all lines of PHG(R 3 R ) are blocked by points of supp K. This argument provides a combinatorial proof for Corollary 2.
Corollary 3. Let R be a chain ring with |R| = p 4r , R/ rad R ∼ = F p 2r , p prime. Then R has a subring S with |S| = p 2r and the projective multiset K with supp K = PHG(S 3 S ) is an irreducible (p 2r (p 2r + p + 1), 1) blocking set in PHG(R 3 R ). Proof. Set q = p 2r . The rings R with |R| = q 2 and R/ rad R ∼ = F q are either In the first case, S is the set of all elements of the form a + bX, where a, b ∈ F √ q . In the second case, S = GR(q, p 2 ). The rest follows by Theorem 2.
Remark 2. The blocking set K from Corollary 3 is not uniquely determined by its parameters. In fact, we can construct a nonisomorphic blocking set H with the same parameters as K and having K (1) = H (1) . We can do this as follows. Take as supp H (1) the points of the Baer subplane of (P (1) , L (1) , J (1) ). Denote the points (resp. the lines) of the Baer subplane by [X i ] (resp. [ℓ i ]) i = 1, . . . , q 2 + q + 1. We can index the points and lines of the Baer subplane in such way that (X i , ℓ i ) ∈ I. Now define the blocking set H by (5) H(P ) = 1 if P ∈ [X i ] ∩ ℓ i for some i ∈ {1, . . . , q 2 + q + 1}, 0 otherwise. Now it is a straightforward check that H is indeed a blocking set and that every point from supp H is incident with a 1-line, i.e. the blocking set is irreducible.
In this section, we consider chain rings R with |R| = q 2 , R/ rad R ∼ = F q for some prime power q. It has been pointed out already that the classification of such chain rings is known. If q = p r there exist exactly r + 1 isomorphism classes of such rings (cf. [23], [29]). These are the Galois rings GR(q 2 , p 2 ) of characteristic p 2 and the truncated skew polynomial rings F q [X; σ]/(X 2 ), σ ∈ Aut F q (cf. [23], [29]). The commutative rings among these are GR(q 2 , p 2 ) and F q [X; id]/(X 2 ).
The blocking sets we described in the previous section have a simple structure in the sense that their image under π (1) is a trivial blocking set in PG(2, q). On the other hand, these blocking sets are not necessarily "sums of lines". By Corollary 1, we have that the minimal size of a blocking set with n = 1 is k = q(q + 1) and the support of such blocking set contains the points of a single line. A natural question is what is the size of the second smallest irreducible blocking set. Corollary 2 gives a family of (q 2 + q + 1, 1)-blocking sets for projective Hjelmslev planes over rings that contain a subring isomorphic to their residue field.
It is possible to construct another class of irreducible blocking sets with k = q 2 + q + 1, n = 1. Consider two lines ℓ and ℓ 0 with ℓ ⌢ ⌣ ℓ 0 and a point X ∈ ℓ \ ℓ 0 . Set It is readily checked that K is indeed a blocking set and that it is irreducible. This construction works for every chain ring R. It turns out that there exist no other irreducible blocking sets of size q 2 + q + 1 in PHG(R 3 R ). Moreover the blocking sets from (6) are the only blocking sets when the characteristic of the ring R is p 2 , i.e. R is the Galois ring GR(q 2 , p 2 ). Theorem 4. Let K be an irreducible (q 2 + q + 1, 1)-blocking set in PHG(R 3 R ), |R| = q 2 , R/ rad R ∼ = F q . Then either supp K is a projective plane of order q or else K is a blocking set of the type described in (6). If R = GR(q 2 , p 2 ), then K is of the type described in (6).
Proof. Let K be an irreducible (q 2 + q + 1, 1)-blocking set in PHG(R 3 R ). By the irreducibility, supp K does not contain a complete line. Assume to get the contradiction |K| ≥ q 2 + q + 2. Thus we have proved that a neighbour class of points in PHG(R 3 R ) has multiplicity 0,1, q, or q + 1. We consider two cases: 1. There exists a neighbour class of points [X] with K([X]) = 0. Consider a neighbour class of lines, [ℓ] say, incident with [X]. In order to block all lines in [ℓ], we need at least q points. On the other hand, K([ℓ]) ≤ q + 1. Therefore, every neighbour class of lines through [X] has multiplicity q except for one class of multiplicity q + 1. Furthermore, using Fact 4 together with the fact that a blocking set with q + 1 or q + 2 points in a projective plane of order q does necessarily contain a line, we get that the q points in every line class through [X] are contained in the same neighbour class. For the line class with q + 1 points we can clame that q of them are neighbours while the (q + 1)-st might be in a different class. The q + 1 neighbour classes of points containing at least q points from supp K must be collinear in the factor plane. Otherwise, there is a line in the factor plane which is . This is a contradiction since we cannot block them by one point outside [X]. This implies that the points from [X] ∩ supp K are collinear and the q lines defined by them are in [ℓ 0 ]. Similarly, we see that the same is true for a class with q + 1 points: q of them are incident with q neighbour lines from [ℓ 0 ] and the (q + 1)-st point is arbitrary. Now consider incidence structure defined on [ℓ 0 ] (cf. Fact 4). It is a isomorphic to the projective plane without a point, which we denote by P ∞ . Define a multiset H by The multiset H is a (q + 1, 1) or (q + 2, 1)-blocking set in a projective plane of order q and hence contains a line. This implies that supp K either contains a complete line, in which case the blocking set is reducible, or else is of the type described in (6 (7). Since H is (q + 1, 1) blocking set in a projective plane of order q the points from the classes [X] with ([X], [ℓ]) ∈ J (1) are collinear. Therefore the lines have multiplicity 1 or q + 1 and the points of supp K together with the (q + 1)-lines form a projective plane of order q. Now we are going to prove that the projective Hjelmslev plane over R = GR(q 2 , p 2 ) cannot contain a subplane isomorphic to a projective plane of order q. Assume otherwise and denote by ∆ a subplane isomorphic to a projective plane of order q contained in Π = PHG(R 3 R ). Without loss of generality ∆ contains the points (1, 0, 0), (1, 1, 0), (1, 0, 1) and (1, 1, 1). Then it contains also (0, 1, 0), (0, 0, 1), and (0, 1, −1). Removing the neighbour class [l], where l is given by x 1 = 0 we get an affine Hjelmslev plane Π isomorphic to AHG(R 2 R ). The points of ∆ \ l form a subplane ∆ isomorphic to an affine plane of order q. Moreover, the points (0, 0), (1, 0), (0, 1) and (1, 1) are in ∆. Denote the two coordinates in Π by y and z. Then the points in ∆ on the line y = 0 are (0, a i ), i = 0, . . . , q − 1, where a 0 = 0, a 1 = 1 and a i ≡ a j (rad R), for i = j, in particular all a i , i = 0 are invertible. The points on the line y = 1 are (1, a i ), i = 0, . . . , q − 1. The intersection point of y = 0 and the line through (1, a j ) parallel to (0, a i ), (1, 0) , is (0, a i + a j ), i.e. the set S = {0, a 1 , a 2 , . . . , a q−1 } is closed under addition. Similarly, all points on the line y = z are (a i , a i ), i = 0, . . . , q − 1. The intersection point of y = 0 and the line through (a j , a j ) parallel to (0, a i ), (1, 1) , i, j = 0, is (0, a i a j ). Hence S is also closed under multiplication and S is a subring of R. But is also a subfield since S ∩ rad R = {a 0 } = {0}. This is a contradiction since charR = p 2 .
Remark 3. The construction from (6) can be generalized. Fix a line ℓ in PHG(R 3 R ) and consider the factor geometry from Fact 4, defined for the line class [ℓ]. First assume that q is even. Fix a hyperoval {ℓ i ∩ [X i ] | i = 1, . . . , q + 1} ∪ P ∞ , where [X i ] are the points incident with [ℓ] in Π (1) . We can choose ℓ to be an external line to this hyperoval and the points X i to be incident (under I) with ℓ. For every s = 0, . . . , q, define the multiset K by The multiset K is an irreducible (q 2 + q + 1 + s, 1)-blocking set if every point ℓ ∩ [X i ] in the factor geometry is incident with at least two external lines to the hyperoval. This is certainly true for q > 2.
In the case q odd, we take {ℓ i ∩ [X i ] | i = 1, . . . , q} ∪ P ∞ to be an oval; ℓ is again an external line to the oval and X i are the same as for q even. The multiset K defined by (8) is again an irreducible (q 2 + q + 1 + s, 1)-blocking set if every point ℓ ∩ [X i ] in the factor geometry is incident with at least two external lines to the oval, which is true for q > 3.

Codes from Blocking Sets
It is well-known that with every multiset K in the projective geometry PHG(R 3 R ), R a chain ring, we can associate a class of isomorphic (left) linear codes over R. Every representative in this class is called a code associated with the multiset K. A code associated with K is is obtained as the module generated by the rows of a matrix with columns the points of supp K taken with the corresponding multiplicities (cf. [11]).
For the sake of simplicity, we consider chain rings R with |R| = q 2 , R/ rad R ∼ = F q . Let K be a multiset in PHG(R 3 R ) with spectrum (a i0,i1 ), and let C = C K be a code associated with K. C can be mapped to a code over a q-letter alphabet (which we take as F q ) in the following way. Let T = {γ 0 , γ 1 , . . . , γ q−1 } be a set of elements of R no two of which are congruent modulo rad R = Rθ. Without loss of generality, γ 0 = 0, γ 1 = 1. Every element r ∈ R is represented uniquely as r = r 0 + r 1 θ, where r 0 , r 1 ∈ T . Set ϕ : T → F q γ → (γ + rad R) σ , where σ : R/ rad R → F q is an isomorphism. Every element r = r 0 + r 1 θ ∈ R can be mapped to a q-tuple over F q by ψ(r) = (r 1 , r 0 ) 1 1 . . . 1 γ 0 γ 1 . . . γ q−1 .
Thus every (right) linear code C over the chain ring R can be mapped to a code ψ(C) over a q-ary alphabet. Generally, ψ(C) is not linear, but it can be made linear if R contains a subring isomorphic to the residue field F q (cf. [11]).
Let K be a multiset of cardinality k in PHG(R 3 R ) and let C be a code associated with K. It is easily verified that a line l produces words of the following nonzero weights in ψ(C): • if l is of type (i 1 , i 2 ), i 1 + i 2 = k, -q 2 − q words of (Hamming) weight qi 1 + (q − 1)(k − i 1 − i 2 ); -q − 1 words of weight q(k − i 1 − i 2 ); • if l is of type (i 1 , i 2 ), i 1 + i 2 = k, q − 1 words of weight qi 1 .