Some binary BCH codes with length \(n = 2^m + 1\). (English) Zbl 1404.94147
Summary: Under research for nearly sixty years, Bose-Chaudhuri-Hocquenghem (BCH) codes have played increasingly important roles in many applications such as communication, data storage and information security. However, the dimension and minimum distance of BCH codes have been seldom solved by now because of their intractable characteristics. The objective of this paper is to study the dimensions of some binary BCH codes with length \(n = 2^m + 1\). Many new techniques are employed to investigate the coset leaders modulo \(n\). For \(m = 2 t + 1, 4 t + 2, 8 t + 4\) and \(m \geq 10\), the first five largest coset leaders modulo \(n\) are determined, and the dimensions of some BCH codes of length \(n\) with designed distance \(\delta > 2^{\lceil \frac{m}{2} \rceil}\) are presented. These new skills and results may be helpful to study other types of cyclic codes over finite fields.
MSC:
| 94B15 | Cyclic codes |
| 11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |
| 14G50 | Applications to coding theory and cryptography of arithmetic geometry |
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