Correction to "On normal and subnormal q-ary codes"

No abstrac

Improved sphere bounds on the covering radius of codes

The sphere bound is a trivial lower bound on K(n,R), the minimal cardinality of any binary code of length n and with covering radius R. By simple arguments it is considerably improved, to K(n,1)⩾2 n/n for n even. A table of lower and upper bounds on K(n,R) for n⩽33, R ⩽10 is include

Bounds on packings and coverings by spheres inq-ary and mixed Hamming spaces

In a recent paper by the same author general improvements on the sphere covering bound for binary covering codes were obtained. In the present work it is shown how the main idea can be exploited to obtain "improved sphere bounds" also for nonbinary codes. We concentrate on generalq-ary codes and on binary/ternary mixed codes. Special attention is paid to the football pool problem; a few new lower bounds are established. Also, it is shown how a similar method yields upper bounds on packing codes

More binary covering codes are normal

It is shown that every optimal binary code with covering radius R=1 is normal. This (parity) proves a conjecture of Cohen, Lobstein, and Sloane (1986). It is also proved that codes with minimal distance 2R or 2R+1 are normal. A generalization of Frankl's construction (1987) of abnormal codes is given

On the non-existence of certain perfect mixed codes

It is shown that if a nontrivial perfect mixed e-code in Q1 × Q2 × × Qn exists, where the Qi are alphabets of size qi, then the qi, e and n satisfy certain divisibility conditions. In particular, all differences qi - qj must be divisible by e + 1

Generalized bounds on binary/ternary mixed packing and covering codes

We derive lower bounds onR-coverings and upper bounds one-packings of . These bounds are generalizations of the bounds for the cases R = 1, e = 1 or t = 0 or b = 0, which were already derived in a paper by the second-named author. The present work is a supplement to that paper. The computed results for t + b = 13 and R, e = 3 are collected in a table

On normal and subnormal q-ary codes

The authors extend to the q-ary case the notions of a normal code, a subnormal code, and the amalgamated direct sum construction, in order to investigate problems related to the covering radius of codes. For example, the authors prove that every nonbinary nontrivial perfect code is absubnormal. They also include some linear-programming lower bounds on ternary codes with covering radius 2 or 3