A modified lower bound for binary codes with covering radius one is derived. Let K/sub q/(n,R) denote the minimum cardinality of a q-ary code with covering radius R. The new bound shows that K/sub 2/(11,1)>or=177 and that K/sub 2/(17,1)>or=7391, improvements of the best lower bounds known. The authors also generalize a known lower bound for binary codes to the case of arbitrary q. For q=3, this simple bound improves the best lower bounds known in several cases. An updated version of a table for K/sub 3/(n,R) is included.>

An infinite series of curves is constructed in order to show that all linear codes can be obtained from curves using Goppa's construction. If conditions are imposed on the degree of the divisor use, then criteria are derived for linear codes to be algebraic-geometric. In particular. the family of q-ary Hamming codes is investigated, and it is proven that only those with redundancy one or two and the binary (7,4,3) code are algebraic-geometric in this sense. For these codes. the authors explicitly give a curve, rational points, and a divisor. It is proven that this triple is in a certain sense unique in the case of the (7,4,3) code.>

Let Q be an alphabet of size q>or=2. The Hamming space Q/sup n/ that consists of all n-tuples of elements of Q is a metric space, provided with the Hamming distance function. A perfect multiple covering (PMC) is a code C in Q/sup n/ such that there exist fixed numbers r and mu with the property that every word in Q/sup n/ is within distance r from exactly mu codewords of C. The authors give a few constructions of PMCs and investigate in detail the problem of determining all possible parameters of PMCs with r=1.>

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.>

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.>

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)>or=2/sup n//n for n even. A table of lower and upper bounds on K(n,R) for n>