The automorphism groups of Reed-Solomon codes

A class of maximum distance separable codes is introduced which includes Reed Solomon codes; extended Reed-Solomon codes, and other cyclic or pseudocyclie MDS codes studied recently. This class of codes, which we call “Cauchy codes” because of the special form of their generator matrices, forms a closed submanifold of dimension 2n - 4 in the k × (n - k)-dimensional algebraic manifold of all MDS codes of length n and dimension k. For every Cauchy code we determine the automorphism group and its underlying permutation group. Far doubly-extended Reed-Solomon codes over GF(q) the permutation group is the semilinear fractional group PΛL(2, q).     

Latin bitrades derived from groups

A Latin bitrade is a pair of partial Latin squares which are disjoint, occupy the same set of non-empty cells, and whose corresponding rows and columns contain the same set of entries. In [A. Drápal, On geometrical structure and construction of Latin trades, Advances in Geometry (in press)] it is shown that a Latin bitrade may be thought of as three derangements of the same set, whose product is the identity and whose cycles pairwise have at most one point in common. By letting a group act on itself by right translation, we show how some Latin bitrades may be derived directly from groups. Properties of Latin bitrades such as homogeneity, minimality (via thinness) and orthogonality may also be encoded succinctly within the group structure. We apply the construction to some well-known groups, constructing previously unknown Latin bitrades. In particular, we show the existence of minimal, k-homogeneous Latin bitrades for each odd k≥3. In some cases these are the smallest known such examples.     

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.     

Planar Eulerian triangulations are equivalent to spherical Latin bitrades

Given a pair of Latin squares, we may remove from both squares those cells that contain the same symbol in corresponding positions. The resulting pair T={P₁,P₂} of partial Latin squares is called a Latin bitrade. The number of filled cells in P₁ is called the size of T. There are at least two natural ways to define the genus of a Latin bitrade; the bitrades of genus 0 are called spherical. We construct a simple bijection between the isomorphism classes of planar Eulerian triangulations on v vertices and the main classes of spherical Latin bitrades of size v−2. Since there exists a fast algorithm (due to Batagelj, Brinkmann and McKay) for generating planar Eulerian triangulations up to isomorphism, our result implies that also spherical Latin bitrades can be generated very efficiently.     

Latin hypercubes and MDS codes

Maximum distance separable (MDS) codes have special properties that give them excellent error correcting capabilities. Counting the number of q-ary MDS codes of length n and distance d, denoted by D_q(n,d)_MDS, is a very hard problem. This paper shows that for d=2, it amounts to counting the number of (n-1)-dimensional Latin hypercubes of order q. Thus, D_q(3,2)_MDS is the number of Latin squares of order q, which is known only for a few values of q. This paper proves constructively that D₃(n,2)_MDS=6·2^n-2.     

Intersection of Isomorphic Linear Codes

Given an (n, k) linear code over GF(q), the intersection of with a codeπ( ), whereπS_n, is an (n, k₁) code, where max{0, 2k−n}k₁k. The intersection problem is to determine which integers in this range are attainable for a given code . We show that, depending on the structure of the generator matrix of the code, some of the values in this range are attainable. As a consequence we give a complete solution to the intersection problem for most of the interesting linear codes, e.g. cyclic codes, Reed–Muller codes, and most MDS codes.     

Affine nonsystematic codes

A perfect binary code C of length n = 2 ^k ? 1 is called affine systematic if there exists a k-dimensional subspace of {0, 1} ⁿ such that the intersection of C and each coset with respect to this subspace is a singleton; otherwise C is called affine nonsystematic. In this article we construct affine nonsystematic codes.     

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 distance Gray codes

A Gray code of size n is a cyclic sequence of all binary words of length n such that two consecutive words differ exactly in one position. We say that the Gray code is a distance code if the Hamming distance between words located at distance k from each other is equal to d. The distance property generalizes the familiar concepts of a locally balanced Gray code. We prove that there are no distance Gray codes with d = 1 for k > 1. Some examples of constructing distance Gray codes are given. For one infinite series of parameters, it is proved that there are no distance Gray codes.     

Ternary self-orthogonal codes of dual distance three and ternary quantum codes of distance three

Ternary self-orthogonal codes with dual distance three and ternary quantum codes of distance three constructed from ternary self-orthogonal codes are discussed in this paper. Firstly, for given code length n ≥ 8, a ternary [n, k]₃ self-orthogonal code with minimal dimension k and dual distance three is constructed. Secondly, for each n ≥ 8, two nested ternary self-orthogonal codes with dual distance two and three are constructed, and consequently ternary quantum code of length n and distance three is constructed via Steane construction. Almost all of these quantum codes constructed via Steane construction are optimal or near optimal, and some of these quantum codes are better than those known before.     

Galois LCD codes over finite fields

In this paper, we generalize the linear complementary dual codes (LCD codes for short) to k-Galois LCD codes, and study them by a uniform method. A necessary and sufficient condition for linear codes to be k-Galois LCD codes is obtained, two classes of k-Galois LCD MDS codes are exhibited. Then, necessary and sufficient conditions for λ-constacyclic codes being k-Galois LCD codes are characterized. Some classes of k-Galois LCD λ-constacyclic MDS codes are constructed. Finally, we study Hermitian LCD λ-constacyclic codes, and present a class of Hermitian LCD λ-constacyclic MDS codes.     

Two-dimensional minimax Latin hypercube designs

We investigate minimax Latin hypercube designs in two dimensions for several distance measures. For the ?^∞-distance we are able to construct minimax Latin hypercube designs of n points, and to determine the minimal covering radius, for all n. For the ?¹-distance we have a lower bound for the covering radius, and a construction of minimax Latin hypercube designs for (infinitely) many values of n. We conjecture that the obtained lower bound is attained, except for a few small (known) values of n. For the ?²-distance we have generated minimax solutions up to n=27 by an exhaustive search method. The latter Latin hypercube designs are included in the website www.spacefillingdesigns.nl.     

On the Number of Latin Squares

We (1) determine the number of Latin rectangles with 11 columns and each possible number of rows, including the Latin squares of order 11, (2) answer some questions of Alter by showing that the number of reduced Latin squares of order n is divisible by f! where f is a particular integer close to (3) provide a formula for the number of Latin squares in terms of permanents of (+1, −1)-matrices, (4) find the extremal values for the number of 1-factorisations of k-regular bipartite graphs on 2n vertices whenever 1 ≤ k ≤ n ≤ 11, (5) show that the proportion of Latin squares with a non-trivial symmetry group tends quickly to zero as the order increases. Received September 3, 2004     

On extremely transitive extended perfect codes

We prove that, for every n = 2 ^k with k ≥ 4, there exist nonequivalent extremely transitive extended perfect codes. A transitive extended perfect code we call extremely transitive if the perfect code obtained from this code by puncturing any coordinate position is not transitive. The classification is given for all extended perfect codes of length 16.     

Nets and their codes

A finitek-net of ordern is an incidence structure ofnk lines andn ² points, with any two lines either meeting once or being parallel, havingk parallel classes ofn lines each, and havingn points on each line. Finite nets are important to the study of finite planes and Latin squares.In this paper finite nets will be studied using the following linear codes: the row space of the incidence vectors of lines, the intersection of this code with its orthogonal, the code generated by differences of parallel lines, and the orthogonal to these codes. Using these codes we are able to recast the Moorhouse conjecture in terms of subcodes of the codes he uses, determine coding-theoretic reasons for a net's being maximal, and generalize a theorem of Assmus and Key which uses codes to classify finite planes of prime order.     

Embedding a Latin square with transversal into a projective space

A Latin square of side n defines in a natural way a finite geometry on 3n points, with three lines of size n and n² lines of size 3. A Latin square of side n with a transversal similarly defines a finite geometry on 3n+1 points, with three lines of size n, n²−n lines of size 3, and n concurrent lines of size 4. A collection of k mutually orthogonal Latin squares defines a geometry on kn points, with k lines of size n and n² lines of size k. Extending the work of Bruen and Colbourn [A.A. Bruen, C.J. Colbourn, Transversal designs in classical planes and spaces, J. Combin. Theory Ser. A 92 (2000) 88-94], we characterise embeddings of these finite geometries into projective spaces over skew fields.     

On even-cycle-free subgraphs of the hypercube

It is shown that the size of any C4k+2-free subgraph of the hypercube Qn, k?3, is o(e(Qn)).     

Hausdorff metric on faces of the <Emphasis Type="Italic">n</Emphasis>-cube

The Hausdorff metric on all faces of the unit n-cube (I ⁿ ) is considered. The notion of a cubant is used; it was introduced as an n-digit quaternary code of a k-dimensional face containing the Cartesian product of k frame unit segments and the face translation code within I ⁿ . The cubants form a semigroup with a unit (monoid) with respect to the given operation of multiplication. A calculation of Hausdorff metric based on the generalization of the Hamming metric for binary codes is considered. The supercomputing issues are discussed.     

MacWilliams Extension Theorem for MDS codes over a vector space alphabet

The MacWilliams Extension Theorem states that each linear isometry of a linear code extends to a monomial map. Unlike the linear codes, in general, nonlinear codes do not have the extension property. In our previous work, in the context of a vector space alphabet, the minimum code length, for which there exists an unextendable code isometry, was determined. In this paper an analogue of the extension theorem for MDS codes is proved. It is shown that for almost all, except 2-dimensional, linear MDS codes over a vector space alphabet the extension property holds. For the case of 2-dimensional MDS codes an improvement of our general result is presented. There are also observed extension properties of near-MDS codes. As an auxiliary result, a new bound on the minimum size of multi-fold partitions of a vector space is obtained.     

Some Combinatorial Aspects of Constructing Bipartite-Graph Codes

We propose geometrical methods for constructing square 01-matrices with the same number n of units in every row and column, and such that any two rows of the matrix contain at most one unit in common. These matrices are equivalent to n-regular bipartite graphs without 4-cycles, and therefore can be used for the construction of efficient bipartite-graph codes such that both the classes of its vertices are associated with local constraints. We significantly extend the region of parameters m, n for which there exist an n-regular bipartite graph with 2m vertices and without 4-cycles. In that way we essentially increase the region of lengths and rates of the corresponding bipartite-graph codes. Many new matrices are either circulant or consist of circulant submatrices: this provides code parity-check matrices consisting of circulant submatrices, and hence quasi-cyclic bipartite-graph codes with simple implementation.