Some upper bounds on the covering radii of linear codes over F-q and their applications
We show that the covering radius R of an [n,k,d] code over F-q is bounded above by R less than or equal to n-n(q)(k, inverted right perpendicular d/q inverted left perpendicular). We strengthen this bound when R less than or equal to d and find conditions under which equality holds. As applications of this and other bounds, we show that all binary linear codes of lengths up to 15, or codimension up to 9, are normal. We also establish the normality of most codes of length 16 and many of codimension 10. These results have applications in the construction of codes that attain t[n,k], the smallest covering radius of any binary linear [n,k] code. We also prove some new results on the amalgamated direct sum ("ADS") construction of Graham and Sloane. We find new conditions assuring normality of the ADS; covering radius 1 less than previously guaranteed for ADS of codes with even norms; good covering codes as ADS without the hypothesis of normality, from concepts p- stable and s- stable; codes with best known covering radii as ADS of two, often cyclic, codes (thus retaining structure so as to be suitable for practical applications).
covering radius normality q-ary codes coset leader griesmer bound amalgamated direct sum construction p-stable s-stable