On the Covering Radius of First-Order Generalized Reed–Muller Codes

On the covering radius of first order generalized Reed-Muller codes

Let q = p, p a prime number. Let B m = Fq[X1, . . . , Xm]/(X q 1 −X1, . . . , X m −Xm); B m actually consists of all the functions from Fq to Fq. We identify B q m with F q q through the application B m → F m q P 7→ (P x∈Fmq For all b ∈ Fq , we denote by 1b the function in B m such that 1b(b) = 1 and for all x 6= b, 1b(x) = 0. The weight |P | of P ∈ B m is Card({x, P (x) 6= 0}). The Hamming dis...

On the covering radius of codes

Designing a good error-correcting code is a packing problem. The corresponding covering problem has received much less attention: now the codewords must be placed so that no vector of the space is very far from the nearest codeword. The two problems are quite different, and with a few exceptions good packings, i.e. codes with a large minimal distance, are usually not especially good coverings. ...

On the Covering Radius Problem for Codes I . Bounds on Normalized Covering Radius

In this two-part paper we introduce the notion of a stable code and give a new upper bound on the normalized covering radius ofa code. The main results are that, for fixed k and large n, the minimal covering radius t[n, k] is realized by a normal code in which all but one of the columns have multiplicity l; hence tin + 2, k] t[n, k] + for sufficiently large n. We also show that codes with n _-<...

On the covering radius of Reed-Muller codes

Cohen, G.D., S.N. Litsyn, On the covering radius of Reed-Muller codes, Discrete Mathematics 106/107 (1992) 147-155. We present lower and upper bounds on the covering radius of Reed-Muller codes, yielding asymptotical improvements on known results. The lower bound is simply the sphere covering one (not very new). The upper bound is derived from a thorough use of a lemma, the ‘essence of Reed-Mul...

On the Covering Radius of Long Goppa Codes