Abstract
It is an interesting problem to determine the parameters of BCH codes, due to their wide applications. In this paper, we determine the dimension and the Bose distance of five families of the narrow-sense primitive BCH codes with the following designed distances:
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(1)
\(\delta _{(a,b)}=a\frac{q^m-1}{q-1}+b\frac{q^m-1}{q^2-1}\), where m is even, \(0\le a \le q-1\), \(1\le b \le q-1\), \(1\le a+b \le q-1\).
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(2)
\(\tilde{\delta }_{(a,b)}=aq^{m-1}+(a+b)q^{m-2}-1\), where m is even, \(0\le a \le q-1\), \(1\le b \le q-1\), \(1\le a+b \le q-1\).
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(3)
\({\delta _{(a,c)}}=a\frac{q^m-1}{q-1}+c\frac{q^{m-1}-1}{q-1}\), where \(m\ge 2\), \(0\le a \le q-1\), \(1\le c \le q-1\), \(1\le a+c \le q-1\).
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(4)
\({\delta }'_{(a,t)}=a\frac{q^{m}-1}{q-1}+\frac{q^{m-1}-1}{q-1}-t\), where \(m\ge 3\), \(0\le a \le q-2\), \(a+2\le t \le q-1\).
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(5)
\({\delta }''_{(a,c,t)}=a\frac{q^{m}-1}{q-1}+c\frac{q^{m-1}-1}{q-1}-t\), where \(m\ge 3\), \(0\le a \le q-3\), \(2\le c \le q-1\), \(1\le a+c \le q-1\), \(1\le t \le c-1\).
Moreover, we obtain the exact parameters of two subfamilies of BCH codes with designed distances \(\bar{\delta }= b\frac{q^m-1}{q^2-1}\) and \(\delta _{(a,t)}= (at+1)\frac{q^m-1}{t(q-1)}\) with even m, \(1\le a \le \big \lfloor \frac{q-2}{t}\big \rfloor \), \(1\le b\le q-1\), \(t>1\) and \(t|(q+1)\). Note that we get the narrow-sense primitive BCH codes with flexible designed distance as to a, b, c, t. Finally, we obtain a lot of the optimal or the best narrow-sense primitive BCH codes.
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Acknowledgements
This research was supported by the National Natural Science Foundation of China under Grant Nos. 61772168, 61972126 and 61802102 and the Anhui Provincial Natural Science Foundation under Grant No. 2008085QA04.
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Pang, B., Zhu, S. & Kai, X. Five families of the narrow-sense primitive BCH codes over finite fields. Des. Codes Cryptogr. 89, 2679–2696 (2021). https://doi.org/10.1007/s10623-021-00942-z
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DOI: https://doi.org/10.1007/s10623-021-00942-z