Asymptotically good sequences of curves and codes

The parameters of a linear block code over the finite field Fq of length n, dimension k and minimum distance d will be denoted by [n, k, d]q or [n, k, d]. The quotient k/n is called the information rate and denoted by R = k/n and the relative minimum distance d/n is denoted by δ. A sequence of codes (Cm|m ∈ N) with parameters [nm, km, dm] over a fixed finite field Fq is called asymptotically good if nm tends to infinity, and dm/nm tends to a non-zero constant δ, and km/nm tends to a non-zero constant R for m→∞. Let Hq(0) = 0 and Hq(x) = xlogq(q − 1)− xlogqx− (1− x)logq(1− x) for 0 < x ≤ (q − 1)/q be the entropy function. Then there exist asymptotically good sequences of codes attaining the the Gilbert-Varshamov (GV) bound