On the binary codes with parameters of triply-shortened 1-perfect codes

We study properties of binary codes with parameters close to the parameters of 1-perfect codes. An arbitrary binary (n?=?2 ^m ? 3, 2 ^n-m-1, 4) code C, i.e., a code with parameters of a triply-shortened extended Hamming code, is a cell of an equitable partition of the n-cube into six cells. An arbitrary binary (n?=?2 ^m ? 4, 2 ^n-m , 3) code D, i.e., a code with parameters of a triply-shortened Hamming code, is a cell of an equitable family (but not a partition) with six cells. As a corollary, the codes C and D are completely semiregular; i.e., the weight distribution of such codes depends only on the minimal and maximal codeword weights and the code parameters. Moreover, if D is self-complementary, then it is completely regular. As an intermediate result, we prove, in terms of distance distributions, a general criterion for a partition of the vertices of a graph (from rather general class of graphs, including the distance-regular graphs) to be equitable.     

A remark on full rank perfect codes

Any full rank perfect 1-error correcting binary code of length n=2^k-1 and with a kernel of dimension n-log(n+1)-m, where m is sufficiently large, may be used to construct a full rank perfect 1-error correcting binary code of length 2^m-1 and with a kernel of dimension n-log(n+1)-k. Especially we may construct full rank perfect 1-error correcting binary codes of length n=2^m-1 and with a kernel of dimension n-log(n+1)-4 for m=6,7,…,10.This result extends known results on the possibilities for the size of a kernel of a full rank perfect code.     

A construction of Gray codes inducing complete graphs

A binary Gray code G(n) of length n, is a list of all 2ⁿn-bit codewords such that successive codewords differ in only one bit position. The sequence of bit positions where the single change occurs when going to the next codeword in G(n), denoted by S(n)?s₁,s₂,…,s_2ⁿ-1, is called the transition sequence of the Gray code G(n). The graph G_G(n) induced by a Gray code G(n) has vertex set {1,2,…,n} and edge set {{s_i,s_i+1}:1?i?2ⁿ-2}. If the first and the last codeword differ only in position s_2ⁿ, the code is cyclic and we extend the graph by two more edges {s_2ⁿ-1,s_2ⁿ} and {s_2ⁿ,s₁}. We solve a problem of Wilmer and Ernst [Graphs induced by Gray codes, Discrete Math. 257 (2002) 585-598] about a construction of an n-bit Gray code inducing the complete graph K_n. The technique used to solve this problem is based on a Gray code construction due to Bakos [A. Ádám, Truth Functions and the Problem of their Realization by Two-Terminal Graphs, Akadémiai Kiadó, Budapest, 1968], and which is presented in D.E. Knuth [The Art of Computer Programming, vol. 4, Addison-Wesley as part of “fascicle” 2, USA, 2005].     

On the structure of some group codes

In this paper, we study the following problem: Which characteristics does a codeC possess when the syntactic monoidsyn(C ^*) of the star closureC ^* ofC is a group? For a codeC, if the syntactic monoidsyn(C ^*) is a group, then we callC a group code. This definition of a group code is different from the one in [1] (see [1], 46–47). Schützenberger had characterized the structure of finite group codes and had proved thatC is a finite group code if and only ifC is a full uniform code (see [5], [8]). Fork-prefix andk-suffix codes withk≥2,k-infix,k-outfix,p-infix,s-infix, right semaphore codes and left semaphore codes, etc., we obtain similar results. It is proved that the above mentioned codes are group codes if and only if they are uniform codes.     

A symmetric Roos bound for linear codes

The van Lint-Wilson AB-method yields a short proof of the Roos bound for the minimum distance of a cyclic code. We use the AB-method to obtain a different bound for the weights of a linear code. In contrast to the Roos bound, the role of the codes A and B in our bound is symmetric. We use the bound to prove the actual minimum distance for a class of dual BCH codes of length q²−1 over Fq. We give cyclic codes [63,38,16] and [65,40,16] over F₈ that are better than the known [63,38,15] and [65,40,15] codes.     

Hadamard matrices and doubly even self-dual error-correcting codes

Let n be an integer with n≡4 (mod 8). For any Hadamard matrices H_n of order n, we give a method to define a doubly even self-dual [2n, n] code C(NH_n). Then we will prove that two Hadamard equivalent matrices define equivalent codes.     

Nonexistence of near-extremal formally self-dual even codes of length divisible by 8

It is a well-known fact that if C is an [n,k,d] formally self-dual even code with n>30, then d?2[n/8]. A formally self-dual (f.s.d.) even code with d=2[n/8] is called near-extremal. Kim and Pless [A note on formally self-dual even codes of length divisible by 8, Finite Fields Appl., available online 13 October 2005.] conjecture that there does not exist a near-extremal f.s.d. (not Type II) code of length n?48 with 8|n. In this paper, we prove that if n?72 and 8|n, then there is no near-extremal f.s.d. even code. This result comes from the negative coefficients of weight enumerators. In addition, we introduce shadow transform in near-extremal f.s.d. even codes. Using this we present some results about the nonexistence of near-extremal f.s.d. even codes with n=48,64.     

On the classification and enumeration of self-dual codes

A complete classification is given of all [22, 11] and [24, 12] binary self-dual codes. For each code we give the order of its group, the number of codes equivalent to it, and its weight distribution. There is a unique [24, 12, 6] self-dual code. Several theorems on the enumeration of self-dual codes are used, including formulas for the number of such codes with minimum distance ? 4, and for the sum of the weight enumerators of all such codes of length n. Selforthogonal codes which are generated by code words of weight 4 are completely characterized.     

On designs and formally self-dual codes

Binary formally self-dual (f.s.d.) even codes are the one type of divisible [2n, n] codes which need not be self-dual. We examine such codes in this paper. On occasion a f.s.d. even [2n, n] code can have a larger minimum distance than a [2n, n] self-dual code. We give many examples of interesting f.s.d even codes. We also obtain a strengthening of the Assmus-Mattson theore. IfC is a f.s.d. extremal code of lengthn2 (mol 8) [n 6 (mod 8)], then the words of a fixed weight inC C hold a 3-design [1-design]. Finally, we show that the extremal f.s.d. codes of lengths 10 and 18 are unique.The author thanks the University of Illinois at Chicago for their hospitality while this work was in progress.This work was supported in part by NSA Grant MDA 904-91-H-0003.     

Directed graph representation of half-rate additive codes over GF(4)

We show that (n, 2 ⁿ ) additive codes over GF(4) can be represented as directed graphs. This generalizes earlier results on self-dual additive codes over GF(4), which correspond to undirected graphs. Graph representation reduces the complexity of code classification, and enables us to classify additive (n, 2 ⁿ ) codes over GF(4) of length up to 7. From this we also derive classifications of isodual and formally self-dual codes. We introduce new constructions of circulant and bordered circulant directed graph codes, and show that these codes will always be isodual. A computer search of all such codes of length up to 26 reveals that these constructions produce many codes of high minimum distance. In particular, we find new near-extremal formally self-dual codes of length 11 and 13, and isodual codes of length 24, 25, and 26 with better minimum distance than the best known self-dual codes.     

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.     

The lengths of Hermitian self-dual extended duadic codes

Duadic codes are a class of cyclic codes that generalize quadratic residue codes from prime to composite lengths. For every prime power q, we characterize integers n such that there is a duadic code of length n over F_q² with a Hermitian self-dual parity-check extension. We derive asymptotic estimates for the number of such n as well as for the number of lengths for which duadic codes exist.     

Cyclic projective planes and binary,extended cyclic self-dual codes

If P is a cyclic projective plane of order n, we give number theoretic conditions on n² + n + 1 so that the binary code of P is contained in a binary cyclic code C whose extension is self-dual. When this containment occurs C does not contain any ovals of P. As a corollary to these conditions we obtain that the extended binary code of a cyclic projective plane of order 2^s is contained in a binary extended cyclic self-dual code if and only if s is odd.     

Symmetric LDPC codes and local testing

Coding theoretic and complexity theoretic considerations naturally lead to the question of generating symmetric, sparse, redundant linear systems. This paper provides a new way of construction with better parameters and new lower bounds.Low Density Parity Check (LDPC) codes are linear codes defined by short constraints (a property essential for local testing of a code). Some of the best (theoretically and practically) used codes are LDPC. Symmetric codes are those in which all coordinates “look the same,” namely there is some transitive group acting on the coordinates which preserves the code. Some of the most commonly used locally testable codes (especially in PCPs and other proof systems), including all “low-degree” codes, are symmetric. Requiring that a symmetric binary code of length n has large (linear or near-linear) distance seems to suggest a “con ict” between 1/rate and density (constraint length). In known constructions, if one is constant, then the other is almost the worst possible - n/poly(logn).Our main positive result simultaneously achieves symmetric low density, constant rate codes generated by a single constraint. We present an explicit construction of a symmetric and transitive binary code of length n, near-linear distance n/(log logn)², of constant rate and with constraints of length (logn)⁴. The construction is in the spirit of Tanner codes, namely the codewords are indexed by the edges of a sparse regular expander graph. The main novelty is in our construction of a transitive (non Abelian!) group acting on these edges which preserves the code. Our construction is one instantiation of a framework we call Cayley Codes developed here, that may be viewed as extending zig-zag product to symmetric codes.Our main negative result is that the parameters obtained above cannot be significantly improved, as long as the acting group is solvable (like the one we use). More specifically, we show that in constant rate and linear distance codes (aka “good” codes) invariant under solvable groups, the density (length of generating constraints) cannot go down to a constant, and is bounded below by (log^(Ω(?)) n)(an Ω(?) iterated logarithm) if the group has a derived series of length ?. This negative result precludes natural local tests with constantly many queries for such solvable “good” codes.     

Multidimensional Latin bitrades

A subset of the n-dimensional k-valued hypercube is a unitrade or united bitrade whenever the size of its intersections with the one-dimensional faces of the hypercube takes only the values 0 and 2. A unitrade is bipartite or Hamiltonian whenever the corresponding subgraph of the hypercube is bipartite or Hamiltonian. The pair of parts of a bipartite unitrade is an n-dimensional Latin bitrade. For the n-dimensional ternary hypercube we determine the number of distinct unitrades and obtain an exponential lower bound on the number of inequivalent Latin bitrades. We list all possible n-dimensional Latin bitrades of size less than 2 ⁿ⁺¹. A subset of the n-dimensional k-valued hypercube is a t-fold MDS code whenever the size of its intersection with each one-dimensional face of the hypercube is exactly t. The symmetric difference of two single MDS codes is a bipartite unitrade. Each component of the corresponding Latin bitrade is a switching component of one of these MDS codes. We study the sizes of the components of MDS codes and the possibility of obtaining Latin bitrades of a size given from MDS codes. Furthermore, each MDS code is shown to embed in a Hamiltonian 2-fold MDS code.     

An Extension Theorem for Linear Codes

One of the first results one meets in coding theory is that a binary linear [n,k,d] code, whose minimum distance is odd, can be extended to an [n + 1, k, d + 1] code. This is one of the few elementary results about binary codes which does not obviously generalise to q-ary codes. The aim of this paper is to give a simple sufficient condition for a q-ary [n, k, d] code to be extendable to an [n + 1, k, d + 1] code. Applications will be given to the construction and classification of good codes, to proving the non- existence of certain codes, and also an application in finite geometry.     

Gray Codes,Loopless Algorithm and Partitions

The generation of efficient Gray codes and combinatorial algorithms that list all the members of a combinatorial object has received a lot of attention in the last few years. Knuth gave a code for the set of all partitions of [n] = {1,2,...,n}. Ruskey presented a modified version of Knuth’s algorithm with distance 2. Ehrlich introduced a looplees algorithm for the set of the partitions of [n]; Ruskey and Savage generalized Ehrlich’s results and introduced two Gray codes for the set of partitions of [n]. In this paper, we give another combinatorial Gray code for the set of the partitions of [n] which differs from the aforementioned Gray codes. Also, we construct a different loopless algorithm for generating the set of all partitions of [n] which gives a constant time between successive partitions in the construction process.      

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).     

Two-weight ternary codes and the equation y2 = 4 × 3a + 13

This paper determines the parameters of all two-weight ternary codes C with the property that the minimum weight in the dual code C^⊥ is at least 4. This yields a characterization of uniformly packed ternary [n, k, 4] codes. The proof rests on finding all integer solutions of the equation y² = 4 × 3^a + 13.     

Some constructions of linearly optimal group codes

We continue here the research on (quasi)group codes over (quasi)group rings. We give some constructions of [n,n-3,3]_q-codes over F_q for n=2q and n=3q. These codes are linearly optimal, i.e. have maximal dimension among linear codes having a given length and distance. Although codes with such parameters are known, our main results state that we can construct such codes as (left) group codes. In the paper we use a construction of Reed-Solomon codes as ideals of the group ring F_qG where G is an elementary abelian group of order q.