Abstract
In this paper we deal with the special class of covering codes consisting ofmultiple coverings of the farthest-off points (MCF). In order to measure the quality of an MCF code, we use a natural extension of the notion of density for ordinary covering codes, that is the $\mu$-density for MCF codes; a generalization of the length function for linear covering codes is also introduced. Our main results consist in a number of upper bounds on such a length function, obtained through explicit constructions, especially for the case of covering radius $R=2$. Akey tool is the possibility of computing the $\mu$-length function in terms of Projective Geometry over finite fields. In fact, linear $(R,\mu )$-MCF codes withparameters $ [n,n-r,d]_{q}R$ have a geometrical counterpart consisting of special subsets of $n$ points in the projective space $PG(n-r-1,q)$. We introduce such objects under the name of $(\rho,\mu)$-saturating sets and we provide a number of example and existence results.Finally, Almost Perfect MCF (APMCF) codes, that is codes for which each word at distance $R$ from the code belongs to {exactly}$\mu $ spheres centered in codewords, are considered and their connections withuniformly packed codes, two-weight codes, and subgroups ofSinger groups are pointed out.
Highlights
For a code C with covering radius R, it is sometimes useful that for every word x at distance R from C there is more than one codeword in the Hamming sphere S(x, R)
Γμ(C, R) ≥ 1 holds and equality is attained precisely when each x ∈ Fnq with d(x, C) = R belongs to exactly μ spheres centered in codewords; if this is the case, C is called an Almost Perfect MCF (APMCF) code; here, AP stands for almost-perfect
If the minimum distance d of C is at least 2R − 1, the best μ-density among linear q-ary codes with same codimension r and covering radius R is achieved by the shortest ones
Summary
For a code C with covering radius R, it is sometimes useful that for every word x at distance R from C there is more than one codeword in the Hamming sphere S(x, R). 3. (R, μ)-MCF linear codes and (R − 1, μ)-saturating sets in projective geometry Let ui be the number of distinct sets of R independent points of S belonging to Ti. we have ui distinct ways of expressing s(x) as a linear combination of R columns of H.
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