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.     

Hypergraphical Codes Arising from Binary Trades

By considering the null space of incidence matrices of trivial designs over GF(2) (the space of 1- (v,k) trades overGF(2)) we obtain families of codes which are optimal for some v and k. Moreover, by generalizing the concept of bond space, the weight enumerator polynomials for these codes are obtained.     

Algebraic-geometry codes, one-point codes, and evaluation codes

One-point codes are those algebraic-geometry codes for which the associated divisor is a non-negative multiple of a single point. Evaluation codes were defined in order to give an algebraic generalization of both one-point algebraic-geometry codes and Reed–Muller codes. Given an -algebra A, an order function on A and given a surjective -morphism of algebras , the ith evaluation code with respect to is defined as the code . In this work it is shown that under a certain hypothesis on the -algebra A, not only any evaluation code is a one-point code, but any sequence of evaluation codes is a sequence of one-point codes. This hypothesis on A is that its field of fractions is a function field over and that A is integrally closed. Moreover, we see that a sequence of algebraic-geometry codes G _i with associated divisors is the sequence of evaluation codes associated to some -algebra A, some order function and some surjective morphism with if and only if it is a sequence of one-point codes.      

On the Key Equation Over a Commutative Ring

We define alternant codes over a commutative ring R and a corresponding key equation. We show that when the ring is a domain, e.g. the p-adic integers, the error-locator polynomial is the unique monic minimal polynomial (equivalently, the unique shortest linear recurrence) of the finite sequence of syndromes and that it can be obtained by Algorithm MR of Norton.WhenR is a local ring, we show that the syndrome sequence may have more than one (monic) minimal polynomial, but that all the minimal polynomials coincide modulo the maximal ideal ofR . We characterise the set of minimal polynomials when R is a Hensel ring. We also apply these results to decoding alternant codes over a local ring R: it is enough to find any monic minimal polynomial over R and to find its roots in the residue field. This gives a decoding algorithm for alternant codes over a finite chain ring, which generalizes and improves a method of Interlando et. al. for BCH and Reed-Solomon codes over a Galois ring.     

On a Class of Traceability Codes

Traceability codes are designed to be used in schemes that protect copyrighted digital data against piracy. The main aim of this paper is to give an answer to a Staddon–Stinson–Wei's problem of the existence of traceability codes with q< w ² and b>q. We provide a large class of these codes constructed by using a new general construction method for q-ary codes.     

On self-orthogonal group ring codes

We obtain structural results about group ring codes over F[G], where F is a finite field of characteristic p > 0 and the Sylow p-subgroup of the Abelian group G is cyclic. As a special case, we characterize cyclic codes over finite fields in the case the length of the code is divisible by the characteristic of the field. By the same approach we study cyclic codes of length m over the ring R = F _q [u], u ^r  = 0 with r  > 0, gcd(m, q) = 1. Finally, we give a construction of quasi-cyclic codes over finite fields.      

Construction of MDS self-dual codes over Galois rings

The purpose of this paper is to construct nontrivial MDS self-dual codes over Galois rings. We consider a building-up construction of self-dual codes over Galois rings as a GF(q)-analogue of (Kim and Lee, J Combin Theory ser A, 105:79–95). We give a necessary and sufficient condition on which the building-up construction holds. We construct MDS self-dual codes of lengths up to 8 over GR(3²,2), GR(3³,2) and GR(3⁴,2), and near-MDS self-dual codes of length 10 over these rings. In a similar manner, over GR(5²,2), GR(5³,2) and GR(7²,2), we construct MDS self-dual codes of lengths up to 10 and near-MDS self-dual codes of length 12. Furthermore, over GR(11²,2) we have MDS self-dual codes of lengths up to 12.      

On the Orthogonality of Geometric Codes

We solve some problems concerning the orthogonality of geometric codes associated with sets of i- and j-dimensional subspaces of PG(n, q). Various applications are found, and we discuss all the interesting cases in small dimensional spaces.     

Spreads, arcs, and multiple wavelength codes

We present several new families of multiple wavelength (2-dimensional) optical orthogonal codes (2D-OOCs) with ideal auto-correlation λ_a=0 (codes with at most one pulse per wavelength). We also provide a construction which yields multiple weight codes. All of our constructions produce codes that are either optimal with respect to the Johnson bound (J-optimal), or are asymptotically optimal and maximal. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q).     

MDS codes over finite principal ideal rings

The purpose of this paper is to study codes over finite principal ideal rings. To do this, we begin with codes over finite chain rings as a natural generalization of codes over Galois rings GR(p ^e , l) (including ). We give sufficient conditions on the existence of MDS codes over finite chain rings and on the existence of self-dual codes over finite chain rings. We also construct MDS self-dual codes over Galois rings GF(2 ^e , l) of length n = 2 ^l for any a ≥ 1 and l ≥ 2. Torsion codes over residue fields of finite chain rings are introduced, and some of their properties are derived. Finally, we describe MDS codes and self-dual codes over finite principal ideal rings by examining codes over their component chain rings, via a generalized Chinese remainder theorem.      

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.     

An Enumeration of Binary Self-Dual Codes of Length 32

A binary self-dual code of length 2k is a (2k, k) binary linear code C with the property that every pair of codewords in C are orthogonal. Two self-dual codes, C ₁ and C ₂, are equivalent if and only if there is a permutation of the coordinates of C ₁ that takes C ₁ into C ₂. The automorphism group of a binary code C is the set of all permutations of the coordinates of C that takes C into itself.The main topic of this paper is the enumeration of inequivalent binary self-dual codes. We have developed algorithms that will take lists of inequivalent small codes and produce lists of larger codes where each inequivalent code occurs only a few times. We have defined a canonical form for codes that allowed us to eliminate the overenumeration. So we have lists of inequivalent binary self-dual codes of length up to 32. The enumeration of the length 32 codes is new. Our algorithm also finds the size of the automorphism group so that we can compute the number of distinct binary self-dual codes for a specific length. This number can also be found by counting and matches our total.     

New Constructions for IPP Codes

Identifiable parent property (IPP) codes are introduced to provide protection against illegal producing of copyrighted digital material. In this paper we consider explicit construction methods for IPP codes by means of recursion techniques. The first method directly constructs IPP codes, whereas the second constructs perfect hash families that are then used to derive IPP codes. In fact, the first construction provides an infinite class of IPP codes having the best known asymptotic behavior. We also prove that this class has a traitor tracing algorithm with a runtime of O(M) in general, where M is the number of codewords.     

Upper and Lower Bounds on Maximum Nonlinearity of n-input m-output Boolean Function

The paper presents lower and upper bounds on the maximumnonlinearity for an n-input m-output Booleanfunction. We show a systematic construction method for a highlynonlinear Boolean function based on binary linear codes whichcontain the first order Reed-Muller code as a subcode. We alsopresent a method to prove the nonexistence of some nonlinearBoolean functions by using nonexistence results on binary linearcodes. Such construction and nonexistence results can be regardedas lower and upper bounds on the maximum nonlinearity. For somen and m, these bounds are tighter than theconventional bounds. The techniques employed here indicate astrong connection between binary linear codes and nonlinear n-input m-output Boolean functions.     

Projective Plane and Planar Quantum Codes

Cellulations of the projective plane RP ^2 define single qubit topological quantum error correcting codes since there is a unique essential cycle in H ₁ (RP ² ;Z ₂ ) . We construct three of the smallest such codes, show they are inequivalent, and identify one of them as Shor's original 9 qubit repetition code. We observe that Shor's code can be constructed in a planar domain and generalize to planar constructions of higher-genus codes for multiple qubits. June 21, 2000. Final version recieved: April 27, 2001.     

On repeated-root multivariable codes over a finite chain ring

In this work we consider repeated-root multivariable codes over a finite chain ring. We show conditions for these codes to be principally generated. We consider a suitable set of generators of the code and compute its minimum distance. As an application we study the relevant example of the generalized Kerdock code in its r-dimensional cyclic version.      

The Automorphism Groups of BCH Codes and of Some Affine-Invariant Codes Over Extension Fields

Affine-invariant codes are extended cyclic codes of length p ^m invariant under the affine-group acting on . This class of codes includes codes of great interest such as extended narrow-sense BCH codes. In recent papers, we classified the automorphism groups of affine-invariant codes berg, bech1. We derive here new results, especially when the alphabet field is an extension field, by expanding our previous tools. In particular we complete our results on BCH codes, giving the automorphism groups of extended narrow-sense BCH codes defined over any extension field.     

New Linear Codes with Covering Radius 2 and Odd Basis

On the way of generalizing recent results by Cock and the second author, it is shown that when the basis q is odd, BCH codes can be lengthened to obtain new codes with covering radius R=2. These constructions (together with a lengthening construction by the first author) give new infinite families of linear covering codes with codimension r=2k+1 (the case q=3, r=4k+1 was considered earlier). New code families with r=4k are also obtained. An updated table of upper bounds on the length function for linear codes with 24, R=2, and q=3,5 is given.     

2-dimensional optical orthogonal codes from singer groups

We present several new families of (Λ×T,w,λ) (2D) wavelength/time optical orthogonal codes (2D-OOCs) with λ=1,2. All families presented are either optimal with respect to the Johnson bound (J-optimal) or are asymptotically optimal. The codes presented have more flexible dimensions and weight than the J-optimal families appearing in the literature. The constructions are based on certain pointsets in finite projective spaces of dimension k over GF(q) denoted PG(k,q). This finite geometries framework gives structure to the codes providing insight. We establish that all 2D-OOCs constructed are in fact maximal (in that no new codeword may be added to the original whereby code cardinality is increased).