The nonexistence of ternary [105,6,68] and [230,6,152] codes

Let [n,k,d]_q-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). In this paper, the nonexistence of [105,6,68]₃ and [230,6,152]₃ codes is proved.     

A new extension theorem for linear codes

For an [n,k,d]_q code with k3, gcd(d,q)=1, the diversity of is defined as the pair (Φ₀,Φ₁) with

All the diversities for [n,k,d]_q codes with k3, d−2 (mod q) such that A_i=0 for all i0,−1,−2 (mod q) are found and characterized with their spectra geometrically, which yields that such codes are extendable for all odd q5. Double extendability is also investigated.     

Euclidean and Hermitian LCD MDS codes

Linear codes with complementary duals (abbreviated LCD) are linear codes whose intersection with their dual is trivial. When they are binary, they play an important role in armoring implementations against side-channel attacks and fault injection attacks. Non-binary LCD codes in characteristic 2 can be transformed into binary LCD codes by expansion. On the other hand, being optimal codes, maximum distance separable codes (abbreviated MDS) are of much interest from many viewpoints due to their theoretical and practical properties. However, little work has been done on LCD MDS codes. In particular, determining the existence of q-ary [n, k] LCD MDS codes for various lengths n and dimensions k is a basic and interesting problem. In this paper, we firstly study the problem of the existence of q-ary [n, k] LCD MDS codes and solve it for the Euclidean case. More specifically, we show that for \(q>3\) there exists a q-ary [n, k] Euclidean LCD MDS code, where \(0\le k \le n\le q+1\), or, \(q=2^{m}\), \(n=q+2\) and \(k= 3 \text { or } q-1\). Secondly, we investigate several constructions of new Euclidean and Hermitian LCD MDS codes. Our main techniques in constructing Euclidean and Hermitian LCD MDS codes use some linear codes with small dimension or codimension, self-orthogonal codes and generalized Reed-Solomon codes.     

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 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.     

Codes of Small Defect

The parameters of a linear code C over GF(q) are given by [n,k,d], where n denotes the length, k the dimension and d the minimum distance of C. The code C is called MDS, or maximum distance separable, if the minimum distance d meets the Singleton bound, i.e. d = n-k+1 Unfortunately, the parameters of an MDS code are severely limited by the size of the field. Thus we look for codes which have minimum distance close to the Singleton bound. Of particular interest is the class of almost MDS codes, i.e. codes for which d=n-k. We will present a condition on the minimum distance of a code to guarantee that the orthogonal code is an almost MDS code. This extends a result of Dodunekov and Landgev Dodunekov. Evaluation of the MacWilliams identities leads to a closed formula for the weight distribution which turns out to be completely determined for almost MDS codes up to one parameter. As a consequence we obtain surprising combinatorial relations in such codes. This leads, among other things, to an answer to a question of Assmus and Mattson 5 on the existence of self-dual [2d,d,d]-codes which have no code words of weight d+1. Actually there are more codes than Assmus and Mattson expected, but the examples which we know are related to the expected ones.     

On the maximality of linear codes

We show that if a linear code admits an extension, then it necessarily admits a linear extension. There are many linear codes that are known to admit no linear extensions. Our result implies that these codes are in fact maximal. We are able to characterize maximal linear (n, k, d) _q -codes as complete (weighted) (n, n − d)-arcs in PG(k − 1, q). At the same time our results sharply limit the possibilities for constructing long non-linear codes. The central ideas to our approach are the Bruen-Silverman model of linear codes, and some well known results on the theory of directions determined by affine point-sets in PG(k, q).      

On a More General Characterisation of Steiner Systems

We introduce [k,d]-sparse geometries of cardinality n, which are natural generalizations of partial Steiner systems PS(t,k;n), with d=2(k−t+1). We will verify whether Steiner systems are characterised in the following way. (*) Let be a [k,2(k−t+1)]-sparse geometry of cardinality n, with k \> t \> 1$$" align="middle" border="0"> . If , then Γ is a S(t,k;n). If (*) holds for fixed parameters t, k and n, then we say S(t,k;n) satisfies, or has, characterisation (*). We could not prove (*) in general, but we prove the Theorems 1, 2, 3 and 4, which state conditions under which (*) is satisfied. Moreover, we verify characterisation (*) for every Steiner system appearing in list of the sporadic Steiner systems of small cardinality, and the list of infinite series of Steiner systems, both mentioned in the latest edition of the book ‘Design Theory’ by T. Beth, D. Jungnickel and H. Lenz. As an interesting application, one can use these results to build (almost) maximal binary codes in the following way. Every [k,d]-sparse geometry is associated with a [k,d]-sparse binary code of the same size (let and link every block with the code word where c_i=1 if and only if the point p_i is a member of B), so one can construct maximal [k,d]-sparse binary codes using (partial) Steiner systems. These [k,d]-sparse codes can then be used as building bricks for binary codes having a bigger variety of weights (the weight of a code word is the sum of its entries).     

Codes that attain minimum distance in every possible direction

The following problem motivated by investigation of databases is studied. Let be a q-ary code of length n with the properties that has minimum distance at least n − k + 1, and for any set of k − 1 coordinates there exist two codewords that agree exactly there. Let f(q, k)be the maximum n for which such a code exists. f(q, k)is bounded by linear functions of k and q, and the exact values for special k and qare determined.      

Some Upper Bounds on the Covering Radii of Linear Codes Over Fq and Their Applications

We show that the covering radius R of an [n,k,d] code over F_q is bounded above by R n-n _q(k, d/q). We strengthen this bound when R d and find conditions under which equality holds.As applications of this and other bounds, we show that all binary linear codes of lengths up to 15, or codimension up to 9, are normal. We also establish the normality of most codes of length 16 and many of codimension 10. These results have applications in the construction of codes that attain t[n,k,/it>], the smallest covering radius of any binary linear [n,k].We also prove some new results on the amalgamated direct sum (ADS) construction of Graham and Sloane. We find new conditions assuring normality of the ADS; covering radius 1 less than previously guaranteed for ADS of codes with even norms; good covering codes as ADS without the hypothesis of normality, from concepts p- stable and s- stable; codes with best known covering radii as ADS of two, often cyclic, codes (thus retaining structure so as to be suitable for practical applications).     

Extending MDS Codes

A q-ary (n, k)-MDS code, linear or not, satisfies n ≤ q + k − 1. A code meeting this bound is said to have maximum length. Using purely combinatorial methods we show that an MDS code with n = q + k − 2 can be uniquely extended to a maximum length code if and only if q is even. This result is best possible in the sense that there is, for example, a non-extendable 4-ary (5, 4)-MDS code. It may be that the proof of our result is as interesting as the result itself. We provide a simple necessary and sufficient condition for code extendability. In future work, this condition might be suitably modified to give an extendability condition for arbitrary (shorter) MDS codes.Received December 1, 2003     

New linear codes over GF(8)

Let [n, k, d; q]-codes be linear codes of length n, dimension k and minimum Hamming distance d over GF(q). Let d₈(n, k) be the maximum possible minimum Hamming distance of a linear [n, k, d; 8]-code for given values of n and k. In this paper, eighteen new linear codes over GF(8) are constructed which improve the table of d₈(n, k) by Brouwer.     

Classification and nonexistence results for linear codes with prescribed minimum distances

Starting from a linear [n, k, d] _q code with dual distance ${d^{\bot}}$ , we may construct an ${[n - d^\bot, k - d^\bot +1,\geq d]_q}$ code with dual distance at least ${\left\lceil\frac{d^\bot}{q}\right\rceil}$ using construction Y ₁. The inverse construction gives a rule for the classification of all [n, k, d] _q codes with dual distance ${d^{\bot}}$ by adding ${d^\bot}$ further columns to the parity check matrices of the smaller codes. Isomorph rejection is applied to guarantee a small search space for this iterative approach. Performing a complete search based on this observation, we are able to prove the nonexistence of linear codes for 16 open parameter sets [n, k, d] _q , q =  2, 3, 4, 5, 7, 8. These results imply 217 new upper bounds in the known tables for the minimum distance of linear codes and establish the exact value in 109 cases.     

On the Extendability of Linear Codes

R. Hill and P. Lizak (1995, in “Proc. IEEE Int. Symposium on Inform. Theory, Whistler, Canada,” pp. 345) proved that every [n, k, d]_q code with gcd(d, q)=1 and with all weights congruent to 0 or d (modulo q) is extendable to an [n+1, k, d+1]_q code with all weights congruent to 0 or d+1 (modulo q). We give another elementary geometrical proof of this theorem, which also yields the uniqueness of the extension.     

Kernels and <Emphasis Type="Italic">p</Emphasis>-Kernels of <Emphasis Type="Italic">p</Emphasis><Superscript><Emphasis Type="Italic">r</Emphasis></Superscript>-ary 1-Perfect Codes

The rank of a q-ary code C is the dimension of the subspace spanned by C. The kernel of a q-ary code C of length n can be defined as the set of all translations leaving C invariant. Some relations between the rank and the dimension of the kernel of q-ary 1-perfect codes, over as well as over the prime field , are established. Q-ary 1-perfect codes of length n=(q^m − 1)/(q − 1) with different kernel dimensions using switching constructions are constructed and some upper and lower bounds for the dimension of the kernel, once the rank is given, are established.Communicated by: I.F. Blake     

A New Approach to the Main Conjecture on Algebraic-Geometric MDS Codes

The Main Conjecture on MDS Codes statesthat for every linear [n, k] MDS code over q, if 1 <k < q, then n q+1,except when q is even and k=3 or k=q-1,in which cases n q +2. Recently, there has beenan attempt to prove the conjecture in the case of algebraic-geometriccodes. The method until now has been to reduce the conjectureto a statement about the arithmetic of the jacobian of the curve,and the conjecture has been successfully proven in this way forelliptic and hyperelliptic curves. We present a new approachto the problem, which depends on the geometry of the curve afteran appropriate embedding. Using algebraic-geometric methods,we then prove the conjecture through this approach in the caseof elliptic curves. In the process, we prove a new result aboutthe maximum number of points in an arc which lies on an ellipticcurve.     

On full-rank perfect codes over finite fields

We propose a construction of full-rank q-ary 1-perfect codes. This is a generalization of the construction of full-rank binary 1-perfect codes by Etzion and Vardy (1994). The properties of the i-components of q-ary Hamming codes are investigated, and the construction of full-rank q-ary 1-perfect codes is based on these properties. The switching construction of 1-perfect codes is generalized to the q-ary case. We propose a generalization of the notion of an i-component of a 1-perfect code and introduce the concept of an (i, σ)-component of a q-ary 1-perfect code. We also present a generalization of the Lindström–Schönheim construction of q-ary 1-perfect codes and provide a lower bound for the number of pairwise distinct q-ary 1-perfect codes of length n.     

The non-existence of Griesmer codes with parameters close to codes of Belov type

Hill and Kolev give a large class of q-ary linear codes meeting the Griesmer bound, which are called codes of Belov type (Hill and Kolev, Chapman Hall/CRC Research Notes in Mathematics 403, pp. 127–152, 1999). In this article, we prove that there are no linear codes meeting the Griesmer bound for values of d close to those for codes of Belov type. So we conclude that the lower bounds of d of codes of Belov type are sharp. We give a large class of length optimal codes with n _q (k, d) = g _q (k, d) + 1.     

On the Achievement of the Griesmer Bound

The main theorem in this paper is that there does not exist an [n,k,d]_q code with d = (k-2)q ^k-1 - (k-1)q^k-2 attaining the Griesmer bound for q k, k=3,4,5 and for q 2k-3, k 6.     

Bounds on Codes Derived by Counting Components in Varshamov Graphs

We are interested in improving the Varshamov bound for finite values of length n and minimum distance d. We employ a counting lemma to this end which we find particularly useful in relation to Varshamov graphs. Since a Varshamov graph consists of components corresponding to low weight vectors in the cosets of a code it is a useful tool when trying to improve the estimates involved in the Varshamov bound. We consider how the graph can be iteratively constructed and using our observations are able to achieve a reduction in the over-counting which occurs. This tightens the lower bound for any choice of parameters n, k, d or q and is not dependent on information such as the weight distribution of a code. This work is taken from the author’s thesis [10]