Lexicographic Order and Linearity

Authors:
A. J. Van Zanten
Affiliations:
Delft University of Technology, Fac. of Technical Mathematics and Informatics, Dept. of Pure Mathematics, P.O. Box 5031, 2600 GA Delft, The Netherlands
Venue:
Designs, Codes and Cryptography
Year:
1997

Citing 4
Cited 2

Lexicographic codes: Error-correcting codes from game theory

IEEE Transactions on Information Theory
Integral lexicographic codes

Discrete Mathematics - Coding Theory
Greedy codes

Journal of Combinatorial Theory Series A
Introduction to Coding Theory

Introduction to Coding Theory

On the Construction of Linear q-ary Lexicodes

Designs, Codes and Cryptography
Bounds on Codes Derived by Counting Components in Varshamov Graphs

Designs, Codes and Cryptography

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Abstract

Let \V bea list of all words of (GF(2))^n, lexicographicallyordered with respect to some basis. Lexicodes are codes constructedfrom \V by applying a greedy algorithm. A shortproof, only based on simple principles from linear algebra, isgiven for the linearity of these codes. The proof holds for anyordered basis, and for any selection criterion, thus generalizingthe results of several authors. An extension of the applied techniqueshows that lexicodes over GF(2^{2^k}) are linearfor a wide choice of bases and for a large class of selectioncriteria. This result generalizes a property of Conway and Sloane.