A Subclass of Binary Goppa Codes With Improved Estimation of the Code Dimension

A new bound for the dimension of binary Goppa codes belonging to a specific subclass is given. This bound improves the well-known lower bound for Goppa codes.     

The New Minimum Distance Bounds of Goppa Codes and Their Decoding

A couple of new lower bounds of the minimum distance of Goppa codes is derived, using an extended field code for a Goppa code which contains the Goppa code as its subfield-subcode. Also presented are procedures for both error-only and error-and-erasure decoding for Goppa codes up to the new lower bounds, based on the Berlekamp-Massey algorithm and the Feng-Tzeng multisequence shift-register synthesis algorithms which have been used for decoding cyclic codes up to the BCH and HT(Hartmann-Tzeng) bounds.     

Proof of Conjectures on the True Dimension of Some Binary Goppa Codes

There is a classical lower bound on the dimension of a binary Goppa code. We survey results on some specific codes whose dimension exceeds this bound, and prove two conjectures on the true dimension of two classes of such codes.Part of this work has been presented at the Sixth International Conference on Finite Fields and Applications, Oaxaca, Mexico, May 2001.AMS classification: 94B65     

Toward the Determination of the Minimum Distance of Two-Point Codes on a Hermitian Curve

This is a first step toward the determination of the parameters of two-point codes on a Hermitian curve. We describe the dimension of such codes and determine the minimum distance of some two-point codes.AMS Classification: 94B27, 14H50, 11T71, 11G20Masaaki Homma - Partially supported by Grant-in-Aid for Scientific Research (15500017), JSPS.Seon Jeong Kim - Partially supported by Korea Research Foundation Grant (KRF-2002-041-C00010).     

On Lower Bounds for the Redundancy of Optimal Codes

The problem of providing bounds on the redundancy of an optimal code for a discrete memoryless source in terms of the probability distribution of the source, has been extensively studied in the literature. The attention has mainly focused on binary codes for the case when the most or the least likely source letter probabilities are known. In this paper we analyze the relationships among tight lower bounds on the redundancy r. Let r _D,i(x) be the tight lower bound on r for D-ary codes in terms of the value x of the i-th most likely source letter probability. We prove that _D,i-1(x) _D,i(x) for all possible x and i. As a consequence, we can bound the redundancy when only the value of a probability (but not its rank) is known. Another consequence is a shorter and simpler proof of a known bound. We also provide some other properties of tight lower bounds. Finally, we determine an achievable lower bound on r in terms of the least likely source letter probability for D 3, generalizing the known bound for the case D = 2.     

Exact Bounds on the Sizes of Covering Codes

A code c is a covering code of X with radius r if every element of X is within Hamming distance r from at least one codeword from c. The minimum size of such a c is denoted by c _r(X). Answering a question of Hämäläinen et al. [10], we show further connections between Turán theory and constant weight covering codes. Our main tool is the theory of supersaturated hypergraphs. In particular, for n > n ₀(r) we give the exact minimum number of Hamming balls of radius r required to cover a Hamming ball of radius r + 2 in {0, 1}ⁿ. We prove that c _r(B _n(0, r + 2)) = _{1 i r + 1} ( ^{(n + i – 1) / (r + 1) 2}) + n / (r + 1) and that the centers of the covering balls B(x, r) can be obtained by taking all pairs in the parts of an (r + 1)-partition of the n-set and by taking the singletons in one of the parts.     

Successive Minimization of the State Complexity of the Self-dual Lattices Using Korkin-Zolotarev Reduced Basis

This work presents a systematic method to successively minimize the state complexity of the self-dual lattices (in the sense that each section of the trellis has the minimum possible number of states fixing its preceding co-ordinates). This is based on representing the lattice on an orthogonal co-ordinate system corresponding to the Gram-Schmidt (GS) vectors of a Korkin-Zolotarev (KZ) reduced basis. As part of the computations, we give expressions for the GS vectors of a KZ basis of the K ₁₂, ₂₄, and BW _n lattices. It is also shown that for the complex representation of the ₂₄ and the BW _n lattices over the set of the Gaussian integers, we have: (i) the corresponding GS vectors are along the standard co-ordinate system, and (ii) the branch complexity at each section of the resulting trellis meets a certain lower bound. This results in a very efficient trellis representation for these lattices over the standard co-ordinate system.     

Roper-Suffridge算子上的偏差定理下界估计的简单证明

Liczberski-Starkov first found a lower bound for ||D(f)|| near the origin, where is the Roper-Suffridge operator on the unit ball Bn in Cn and F is a normalized convex function on the unit disk. Later, Liczberski-Starkov and Hamada-Kohr proved the lower bound holds on the whole unit ball using a complex computation. Here we provide a rather short and easy proof for the lower bound. Similarly, when F is a normalized starlike function on the unit disk, a lower bound of ||D(f)|| is obtained again.     

Some Upper Bounds on the Covering Radii of Linear Codes Over Fq and Their Applications

We show that the covering radius R of an [n,k,d] code over F_q is bounded above by R n-n _q(k, d/q). We strengthen this bound when R 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,/it>], the smallest covering radius of any binary linear [n,k].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).