Optimal ternary linear codes

Let n _q (k, d) denote the smallest value of n for which there exists a linear [n, k, d]-code over GF(q). An [n, k, d]-code whose length is equal to n _q (k, d) is called optimal. The problem of finding n _q (k, d)has received much attention for the case q = 2. We generalize several results to the case of an arbitrary prime power q as well as introducing new results and a detailed methodology to enable the problem to be tackled over any finite field.In particular, we study the problem with q = 3 and determine n ₃(k, d) for all d when k 4, and n ₃(5, d) for all but 30 values of d.     

On the minimum length of ternary linear codes

From the geometrical point of view, we prove that [g ₃(6, d) + 1, 6, d]₃ codes exist for d = 118–123, 283–297 and that [g ₃(6, d), 6, d]₃ codes for d = 100, 341, 342 and [g ₃(6, d) + 1, 6, d]₃ codes for d = 130, 131, 132 do not exist, where ${g_3(k,\,d)=\sum_{i=0}^{k-1}\left\lceil d/3^i \right\rceil}$ . These determine the exact value of n ₃(6, d) for d = 100, 118–123, 130, 131, 132, 283–297, 341, 342, where n _q (k, d) is the minimum length n for which an [n, k, d] _q code exists.     

On the nonexistence of ternary linear codes attaining the Griesmer bound

Designs, Codes and Cryptography - An $$[n,k,d]_q$$ code is a linear code of length n, dimension k and minimum weight d over the field of order q. It is known that the Griesmer bound is attained for...     

Construction of binary and ternary self-orthogonal linear codes

We construct new binary and ternary self-orthogonal linear codes. In order to do this we use an equivalence between the existence of a self-orthogonal linear code with a prescribed minimum distance and the existence of a solution of a certain system of Diophantine linear equations. To reduce the size of the system of equations we restrict the search for solutions to solutions with special symmetry given by matrix groups. Using this method we found at least six new distance-optimal codes, which are all self-orthogonal.     

On optimal superimposed codes

A (w,r) cover‐free family is a family of subsets of a finite set such that no intersection of w members of the family is covered by a union of r others. A binary (w,r) superimposed code is the incidence matrix of such a family. Such a family also arises in cryptography as a concept of key distribution patterns. In this paper, we develop a method of constructing superimposed codes and prove that some superimposed codes constructed in this way are optimal. © 2003 Wiley Periodicals, Inc. J Combin Designs 12: 79–71, 2004.     

On diameter perfect constant-weight ternary codes

From cosets of binary Hamming codes we construct diameter perfect constant-weight ternary codes with weight n-1 (where n is the code length) and distances 3 and 5. The class of distance 5 codes has parameters unknown before.     

On non-projective normal surfaces

In this note we construct examples of non-projective normal proper algebraic surfaces and discuss the somewhat pathological behaviour of their Neron–Severi group. Our surfaces are birational to the product of a projective line and a curve of higher genus. Received: 29 January 1999     

Several classes of optimal ternary cyclic codes with minimal distance four

In this paper, by analyzing the solutions of certain equations over ${\mathbb{F}}_{3^m}$, we present four classes of optimal ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$. It is shown that some recent work on this class of optimal ternary cyclic codes are special cases of our results.     

Construction of optimal ternary constant weight codes via Bhaskar Rao designs

In this note, we consider a construction for optimal ternary constant weight codes (CWCs) via Bhaskar Rao designs (BRDs). The known existence results for BRDs are employed to generate many new optimal nonlinear ternary CWCs with constant weight 4 and minimum Hamming distance 5.     

New classes of optimal ternary cyclic codes with minimum distance four

Cyclic code is an interesting topic in coding theory and communication systems. In this paper, we investigate the ternary cyclic codes with parameters $[3^m-1,3^m-1-2m,4]$ based on some results proposed by Ding and Helleseth in 2013. Six new classes of optimal ternary cyclic codes are presented by determining the solutions of certain equations over ${\mathbb{F}}_{3^m}$.     

Three-weight ternary linear codes from a family of cyclic difference sets

Linear codes with a few weights have applications in data storage systems, secret sharing schemes, and authentication codes. Recently, Ding (IEEE Trans. Inf. Theory 61(6):3265–3275, 2015) proposed a class of ternary linear codes with three weights from a family of cyclic difference sets in \(({\mathbb {F}}_{3^m}^*/{\mathbb {F}}_{3}^*,\times )\), where \(m=3k\) and k is odd. One objective of this paper is to construct ternary linear codes with three weights from cyclic difference sets in \(({\mathbb {F}}_{3^m}^*/{\mathbb {F}}_{3}^*,\times )\) derived from the Helleseth–Gong functions. This construction works for any positive integer \(m=sk\) with an odd factor \(s\ge 3\), and thus leads to three-weight ternary linear codes with more flexible parameters than earlier ones mentioned above. Another objective of this paper is to determine the weight distribution of the proposed linear codes.     

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).