On the covering radius of cyclic linear codes and arithmetic codes

The problem of finding the covering radius and minimum distance of algebraic and arithmetic codes is shown to be related to Waring's problem in a finite field and to the theory of cyclotomic numbers. The methods developed lead to new results for the covering radius of certain t-error-correcting BCH codes. Further, new results are given for the covering radius and the minimum distance of some classes of arithmetic codes generated by prime numbers.

[1]  Elwyn R. Berlekamp,et al.  On the inherent intractability of certain coding problems (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[2]  Joseph J. Liang,et al.  Weak radix representation and cyclic codes over euclidean domains , 1976 .

[3]  Charles Small,et al.  Waring's Problem \opertornameMod n , 1977 .

[4]  Toby Berger,et al.  Complete decoding of triple-error-correcting binary BCH codes , 1976, IEEE Trans. Inf. Theory.

[5]  Chao-Kai Liu,et al.  Error-Correcting-Codes in Computer Arithmetic , 1972 .

[6]  Philippe Delsarte,et al.  Four Fundamental Parameters of a Code and Their Combinatorial Significance , 1973, Inf. Control..

[7]  James L. Massey,et al.  Error-correcting codes in computer arithmetic. , 1972 .

[8]  Charles Small,et al.  Solution of Waring's Problem Mod n , 1977 .

[9]  T. Storer.  Cyclotomy and difference sets , 1967 .

[10]  Neal Zierler,et al.  Two-Error Correcting Bose-Chaudhuri Codes are Quasi-Perfect , 1960, Inf. Control..

[11]  Charles Small,et al.  Waring's Problem Modn , 1977 .

[12]  L. Carlitz,et al.  Bounds for exponential sums , 1957 .

[13]  Tor Helleseth.  All binary 3-error-correcting BCH codes of length 2m-i have covering radius 5 (Corresp.) , 1978, IEEE Trans. Inf. Theory.

[14]  James L. Massey,et al.  Review of 'Error-Correcting Codes, 2nd edn.' (Peterson, W. W., and Weldon, E. J., Jr.; 1972) , 1973, IEEE Trans. Inf. Theory.

[15]  Charles Small,et al.  Solution of Waring's Problem \opertornameMod n , 1977 .

[16]  W. Edwin Clark,et al.  On modular weight and cyclic nonadjacent forms for arithmetic codes (Corresp.) , 1974, IEEE Trans. Inf. Theory.

[17]  Tadao Kasami,et al.  Some Remarks on BCH Bounds and Minimum Weights of Binary Primitive BCH Codes , 1969, IEEE Trans. Inf. Theory.

[18]  F. MacWilliams,et al.  The Theory of Error-Correcting Codes , 1977 .

[19]  W. W. Peterson,et al.  Error-Correcting Codes. , 1962 .

[20]  M. M. Dodson,et al.  A note on Waring's problem in GF (p) , 1976 .

[21]  Torleiv Kløve,et al.  The weight distribution of irreducible cyclic codes with block lengths n1((q1-1)/N) , 1977, Discret. Math..

[22]  Robert J. McEliece,et al.  Weights of Irreducible Cyclic Codes , 1972, Inf. Control..

[23]  A. E. Western,et al.  Tables of indices and primitive roots , 1968 .