On the admissible families of components of hamming codes

In this paper, the properties of the i-components of Hamming codes are described. We suggest constructions of the admissible families of components of Hamming codes. Each q-ary code of length m and minimum distance 5 (for q = 3, the minimum distance is 3) is shown to embed in a q-ary 1-perfect code of length n = (q ^m − 1)/(q − 1). Moreover, each binary code of length m+k and minimum distance 3k + 3 embeds in a binary 1-perfect code of length n = 2 ^m − 1.     

Multiple blocking sets in finite projective spaces and improvements to the Griesmer bound for linear codes

Belov, Logachev and Sandimirov construct linear codes of minimum distance d for roughly 1/q ^k/2 of the values of d < q ^k-1. In this article we shall prove that, for q = p prime and roughly \frac38{\frac{3}{8}}-th’s of the values of d < q ^k-1, there is no linear code meeting the Griesmer bound. This result uses Blokhuis’ theorem on the size of a t-fold blocking set in PG(2, p), p prime, which we generalise to higher dimensions. We also give more general lower bounds on the size of a t-fold blocking set in PG(δ, q), for arbitrary q and δ ≥ 3. It is known that from a linear code of dimension k with minimum distance d < q ^k-1 that meets the Griesmer bound one can construct a t-fold blocking set of PG(k−1, q). Here, we calculate explicit formulas relating t and d. Finally we show, using the generalised version of Blokhuis’ theorem, that nearly all linear codes over \mathbb F_p{{\mathbb F}_p} of dimension k with minimum distance d < q ^k-1, which meet the Griesmer bound, have codewords of weight at least d + p in subcodes, which contain codewords satisfying certain hypotheses on their supports.     

On codes with covering radius 1 and minimum distance 2

We show that a code C of length n over an alphabet Q of size q with minimum distance 2 and covering radius 1 satisfies |C| ≥ qⁿ⁻¹/(n − 1). For the special case n = q = 4 the smallest known example has |C| = 31. We give a construction for such a code C with |C| = 28.     

Optimal constant weight covering codes and nonuniform group divisible 3-designs with block size four

Let K _q (n, w, t, d) be the minimum size of a code over Z _q of length n, constant weight w, such that every word with weight t is within Hamming distance d of at least one codeword. In this article, we determine K _q (n, 4, 3, 1) for all n ≥ 4, q = 3, 4 or q = 2 ^m  + 1 with m ≥ 2, leaving the only case (q, n) = (3, 5) in doubt. Our construction method is mainly based on the auxiliary designs, H-frames, which play a crucial role in the recursive constructions of group divisible 3-designs similar to that of candelabra systems in the constructions of 3-wise balanced designs. As an application of this approach, several new infinite classes of nonuniform group divisible 3-designs with block size four are also constructed.     

An upper bound for the minimum weight of dual codes of Figueroa planes

In this note we give an explicit construction for words of weight 2q³ - q² - q in the dual p-ary code of the Figueroa plane of order q³, where q > 2 is any power of the prime p. When p is odd this then allows us, for the Figueroa planes, to improve on the previously known upper bound of 2q³ for the minimum weight of the dual p-ary code of any plane of order q³. The construction is the same as one that applies to desarguesian planes of order q³ as described in [3].     

Artin–Schreier curves and weights of two-dimensional cyclic codes

Let be the finite field with q elements of characteristic p, be the extension of degree m>1 and f(x) be a polynomial over . The maximum number of affine -rational points that a curve of the form y^q−y=f(x) can have is q^m+1. We determine a necessary and sufficient condition for such a curve to achieve this maximum number. Then we study the weights of two-dimensional (2-D) cyclic codes. For this, we give a trace representation of the codes starting with the zeros of the dual 2-D cyclic code. This leads to a relation between the weights of codewords and a family of Artin–Schreier curves. We give a lower bound on the minimum distance for a large class of 2-D cyclic codes. Then we look at some special classes that are not covered by our main result and obtain similar minimum distance bounds.     

Minimal cyclic codes of length pq

Explicit expressions for all the 3n+2 primitive idempotents in the ring R_pⁿq=GF(ℓ)[x]/(x^pnq−1), where p,q,ℓ are distinct odd primes, ℓ is a primitive root modulo pⁿ and q both, , are obtained. The dimension, generating polynomials and the minimum distance of the minimal cyclic codes of length pⁿq over GF(ℓ) are also discussed.     

Minimum distance of orthogonal line-Grassmann codes in even characteristic

In this paper we determine the minimum distance of orthogonal line-Grassmann codes for q even. The case q odd was solved in [3]. For $n\ne 3$ we also determine their second smallest distance. Furthermore, we show that for q even all minimum weight codewords are equivalent and that the symplectic line-Grassmann codes are proper subcodes of codimension 2n of the orthogonal ones.     

On (q 2 + q + 2, q + 2)-arcs in the Projective Plane PG(2, q)

A (k,n)-arc in PG(2,q) is usually defined to be a set of k points in the plane such that some line meets in n points but such that no line meets in more than n points. There is an extensive literature on the topic of (k,n)-arcs. Here we keep the same definition but allow to be a multiset, that is, permit to contain multiple points. The case k=q ²+q+2 is of interest because it is the first value of k for which a (k,n)-arc must be a multiset. The problem of classifying (q ²+q+2,q+2)-arcs is of importance in coding theory, since it is equivalent to classifying 3-dimensional q-ary error-correcting codes of length q ²+q+2 and minimum distance q ². Indeed, it was the coding theory problem which provided the initial motivation for our study. It turns out that such arcs are surprisingly rich in geometric structure. Here we construct several families of (q ²+q+2,q+2)-arcs as well as obtain some bounds and non-existence results. A complete classification of such arcs seems to be a difficult problem.     

Classifying Subspaces of Hamming Spaces

A linear code in F ⁿ _q with dimension k and minimum distance at least d is called an [n, k, d] _q code. We here consider the problem of classifying all [n, k, d] _q codes given n, k, d, and q. In other words, given the Hamming space F ⁿ _q and a dimension k, we classify all k-dimensional subspaces of the Hamming space with minimum distance at least d. Our classification is an iterative procedure where equivalent codes are identified by mapping the code equivalence problem into the graph isomorphism problem, which is solved using the program nauty. For d = 3, the classification is explicitly carried out for binary codes of length n 14, ternary codes of length n 11, and quaternary codes of length n 10.     

On a set of lines of PG(3, q) corresponding to a maximal cap contained in the Klein quadric of PG (5, q)

The problem is considered of constructing a maximal set of lines, with no three in a pencil, in the finite projective geometry PG(3, q) of three dimensions over GF(q). (A pencil is the set of q+1 lines in a plane and passing through a point.) It is found that an orbit of lines of a Singer cycle of PG(3, q) gives a set of size q ³ + q ² + q + 1 which is definitely maximal in the case of q odd. A (q ³ + q ² + q + 1)-cap contained in the hyperbolic (or Klein) quadric of PG(5, q) also comes from the construction. (A k-cap is a set of k points with no three in a line.) This is generalized to give direct constructions of caps in quadrics in PG(5, q). For q odd and greater than 3 these appear to be the largest caps known in PG(5, q). In particular it is shown how to construct directly a large cap contained in the Klein quadric, given an ovoid skew to an elliptic quadric of PG(3, q). Sometimes the cap is also contained in an elliptic quadric of PG(5, q) and this leads to a set of q ³ + q ² + q + 1 lines of PG(3,q ²) contained in the non-singular Hermitian surface such that no three lines pass through a point. These constructions can often be applied to real and complex spaces.     

On the construction of [q 4 + q 2 − q, 5,q 4 − q 3 + q 2 − 2q; q]-codes meeting the Griesmer bound

It is unknown whether or not there exists an [87, 5, 57; 3]-code. Such a code would meet the Griesmer bound. The purpose of this paper is to give a constructive proof of the existence of [q ⁴ + q ² – q, 5, q ⁴ – q ³ + q ² – 2q; q]-codes for any prime power q 3. As a special case, it is shown that there exists an [87, 5, 57; 3]-code with weight enumerator 1 + 156z ³⁷ + 82z ⁶⁰ + 2z ⁶³ + 2z ⁷⁸. The new construction settles an open problem due to Hill and Newton [10].     

Sublines of Prime Order Contained in the Set of Internal Points of a Conic

In [2] it was shown that if q ≥ 4n²−8n+2 then there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qⁿ), q odd, n≥ 3. In this article we improve this bound in the case where q is prime to , and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in PG(3,qⁿ), ovoids of Q(4,qⁿ), rank two commutative semifields, and eggs in PG(4n−1,q). AMS Classification:11T06, 05B25, 05E12, 51E15     

On the Minimum Length of some Linear Codes of Dimension 5

In this paper, we shall prove that the minimum length n_q(5,d) is equal to g_q(5,d) +1 for q⁴−2q²−2q+1≤ d≤ q⁴ − 2q² − q and 2q⁴ − 2q³ − q² − 2q+1 ≤ d ≤ 2q⁴−2q³−q²−q, where g_q(5,d) means the Griesmer bound . Communicated by: J.D. Key     

On the embedding of constant-weight codes into perfect codes

We show that each q-ary constant-weight code of weight 3, minimum distance 4, and length m embeds in a q-ary 1-perfect code of length n = (q ^m ? 1)/(q ? 1).     

Slight improvements of the Singleton bound

The problem of determining A_q(n,d), the maximum cardinality of a q-ary code of length n with minimum distance at least d, is considered in some cases where corresponding MDS codes do not exist. Slight improvements of the Singleton bound are given, including A_q(q+2,q)?q³-3 if q is odd, A₅(7,5)?5³-4 and A₁₆(18,15)?18⁴-4.     

A unified construction for c*· c- and L · L*-geometries in projective spaces

We present two new constructions for c^* · c-geometries. The first provides, for each even prime powerq, a flag-transitive c^* · c-geometry of orderq–1 that is embedded in the projective space PG(3,q) and which is related with the Cameron-Fisher extended grids of odd type. The second construction is valid independently of the parity ofq. Forq even, it produces the same geometry as the first construction, and forq odd, two geometries related with some extended grids constructed by Meixner and Pasini.Next, by using some complementary models for c^* and L in a projective plane, we derive from our construction a new family of L · L^*-geometries embedded in PG(3,q). Forq even, these geometries are flag-transitive.     

On arcs sharing the maximum number of points with ovals in cyclic affine planes of odd order

The sporadic complete 12‐arc in PG(2, 13) contains eight points from a conic. In PG(2,q) with q>13 odd, all known complete k‐arcs sharing exactly ½(q+3) points with a conic 𝒞 have size at most ½(q+3)+2, with only two exceptions, both due to Pellegrino, which are complete (½(q+3)+3) arcs, one in PG(2, 19) and another in PG(2, 43). Here, three further exceptions are exhibited, namely a complete (½(q+3)+4)‐arc in PG(2, 17), and two complete (½(q+3)+3)‐arcs, one in PG(2, 27) and another in PG(2, 59). The main result is Theorem 6.1 which shows the existence of a (½(q^r+3)+3)‐arc in PG(2,q^r) with r odd and q≡3 (mod 4) sharing ½(q^r+3) points with a conic, whenever PG(2,q) has a (½(q^r+3)+3)‐arc sharing ½(q^r+3) points with a conic. A survey of results for smaller q obtained with the use of the MAGMA package is also presented. © 2009 Wiley Periodicals, Inc. J Combin Designs 18: 25–47, 2010     

The Length of Primitive BCH Codes with Minimal Covering Radius

It is proved that the covering radius of a primitive binary BCH code of length q-1 and designed distance 2t+1, where is exactly 2t-1 (the minimum value possible). The bound for q is significantly lower than the one obtained by O. Moreno and C. J. Moreno [9].     

A Characterization of the Complement of a Hyperbolic Quadric in PG (3, q), for q Odd

The incidence structure NQ⁺(3, q) has points the points not on a non-degenerate hyperbolic quadric Q⁺(3, q) in PG(3, q), and its lines are the lines of PG(3, q) not containing a point of Q⁺(3, q). It is easy to show that NQ⁺(3, q) is a partial linear space of order (q, q(q−1)/2). If q is odd, then moreover NQ⁺(3, q) satisfies the property that for each non-incident point line pair (x,L), there are either (q−1)/2 or (q+1)/2 points incident with L that are collinear with x. A partial linear space of order (s, t) satisfying this property is called a ((q−1)/2,(q+1)/2)-geometry. In this paper, we will prove the following characterization of NQ⁺(3,q). Let S be a ((q−1)/2,(q+1)/2)-geometry fully embedded in PG(n, q), for q odd and q>3. Then S = NQ⁺(3, q).