Improved Asymptotic Bounds For Algebraic-Geometric Codes

For any prime power q, letα_q(δ) be the standard function in the asymptotic theory of coding, i.e.,α_q(δ) is the largest asymptotic (relative) information rate that can be achieved for a given asymptotic relative minimum distanceδof q-ary codes. A central problem in the asymptotic theory of coding is to find lower bounds ofα_q(δ) for 0 <δ< (q-1)/q. A known lower bound ofα_q(δ) is the Gilbert-Varshamov (GV) bound: 1- H_q(δ), where H_q(δ) denotes the q-ary entropy function. Tsfasman and others made a breakthrough in the coding theory in 1982. They improved the GV bound by using Goppa's construction based on curves over finite fields of specific order and found the Tsfasman- Vladut-Zink (TVZ) bound: 1-δ- A(q)^-1. After that, algebraic-geometry codes became a hot research topic. Very recently, the TVZ bound was improved by Elkies, Xing, Niederreiter, Ozbudak, Stichtenoth, Maharaj et al.In this thesis, we present a new improvement on the lower bound ofα_q(δ) by using Niederreiter and Qzbudak's method [13]. The key point is that here we introduce a set U(n, s; r₀, r₁) instead of the set U(n, s, w) constructed by Niederreiter and Ozbudak. By employing the second derivative of functions defined on n rational places of functionfields, we are able to introduce two parameters x and y to estimate the lower bound ofα_q(δ). We show that the bound 1-δ-A(q)^-1+log_q(1+2/q³)+log_q(1+ (q-1)/q⁶) can be achieved for certain values of q andδ. We also show that the construction in [13] is a special case of ours.