Juriaan Simonis

Almost Affine Codes

An almost affine code is a code C for which the size of all codes obtained by multiple puncturing of C is a power of the alphabet size. Essentially, almost affine codes are the same as ideal perfect secret haring schemes or partial affine geometries. The present paper explores these interrelations, gives short proofs of known and new results, and derives some properties of the distance distribution of almost affine codes.

The Effective Length of Subcodes

The effective length distribution (support weight distribution in Kløves terminology [3]) of a linear code is the set of integers indicating the number of subcodes of the same dimension and the same effective length. A simple proof is presented to MacWilliams type identities relating the effective length distributions of a linear code and its dual.

Metrics generated by families of subspaces

A new family of metrics is introduced. Each of these is defined by a spanning set F of linear subspaces of a finite vector space. The norm of a vector is defined as the size of a minimal subset of F whose span contains this vector. Some examples and applications are presented. A-class of Varshamov-Gilbert bound based F-metrics is introduced. Connections with combinatorial metrics are discussed.

MacWilliams identities and coordinate partitions

Abstract Any partition of the coordinate set of a linear code is shown to correspond to a set of generalized MacWilliams identities. Thus, a well-chosen partition yields a promising method to settle existence and uniqueness problems. A short proof of an extension of the Assmus-Mattson theorem is given. In the nonlinear case, a generalization of the Delsarte inequalities is obtained.

The [18,9,6] code is unique

Abstract The paper contains a proof that all binary linear [18, 9, 6] codes are equivalent to the extended quadratic residue code of length 18. In addition, a complete description of the words and cosets of this code is given in terms of the special projective structure of the projective line over F 17 .

Adding a parity-check bit

The article gives a new condition for a q-ary linear code of length n and minimum distance d to be extendable to a code of the same dimension, length n+1, and minimum distance >d.

Some new results for optimal ternary linear codes

Let d/sub 3/(n,k) be the maximum possible minimum Hamming distance of a ternary [n,k,d]-code for given values of n and k. We describe a package for code extension and use this to prove some new exact values of d/sub 3/(n,k). Moreover, we classify the ternary [n,k,d/sub 3/(n,k)]-codes for some values of n and k.

On Generator Matrices of Codes

The if class of the q-ary linear codes of given length, dimension and minimum weight is nonempty, it is shown to contain a code whose generator matrix consists of words of minimum weight.

Some New Results on Optimal Codes Over F 5

We present some results on almost maximum distance separable (AMDS) codes and Griesmer codes of dimension 4 over over the field of order 5. We prove that no AMDS code of length 13 and minimum distance 5 exists, and we give a classification of some AMDS codes. Moreover, we classify the projective strongly optimal Griesmer codes over F5 of dimension 4 for some values of the minimum distance.

Optimal Subcodes of Second Order Reed-Muller Codes andMaximal Linear Spaces of Bivectors of Maximal Rank

AbstractThere are exactlytwo non-equivalent [32,11,12]-codes in the binaryReed-Muller code