Generalization of trace codes to places of higher degree

2020/02/29 by Nupur Patanker, Sanjay Kumar Singh, Patanker, Nupur +1
Computer Science · Engineering · #14H05 #94B05 #94B27 #Algebraic Geometry (math.AG) #Cellular Automata and Applications #Coding theory and cryptography #FOS: Computer and information sciences #FOS: Mathematics #Information Theory (cs.IT) #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.2003.00145

openalex publication_date 2020/02/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note, we give a construction of codes on algebraic function field F/ \mathbbF_q using places of F (not necessarily of degree one) and trace functions from various extensions of \mathbbF_q. This is a generalization of trace code of geometric Goppa codes to higher degree places. We compute a bound on the dimension of this code. Furthermore, we give a condition under which we get exact dimension of the code. We also determine a bound on the minimum distance of this code in terms of B_r(F) ( the number of places of degree r in F), 1 ≤ r < ∞. Few quasi-cyclic codes over \mathbbF_p are also obtained as examples of these codes.

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