Complementary information set codes (CIS codes) over a finite field GF(p) are closely connected to correlation-immune functions over GF(p), which are important cryptographic functions, where p is an odd prime. Using our CIS codes over GF(p) of minimum weight d+ 1 , we can obtain p-ary correlation-immune function of strength d. We find an efficient method for constructing CIS codes over GF(p). We also find a criterion for checking equivalence of CIS codes over GF(p). We complete the classification of all inequivalent CIS codes over GF(p) of lengths up to 8 for p= 3 , 5 , 7 using our construction and criterion. We also find their weight enumerators and the order of their automorphism groups. The class of CIS codes over GF(p) includes self-dual codes over GF(p) as its subclass, and some CIS codes are formally self-dual codes as well; we sort out our classification results. Furthermore, we show that long CIS codes over GF(p) meet the Gilbert–Vashamov bound.
Bibliographical noteFunding Information:
The authors are grateful to anonymous referees and a handling editor for their careful review and constructive suggestions for improvement of our manuscript. The authors were supported by the National Research Foundation of Korea (NRF) Grant founded by the Korea government (MEST) (2014-002731), the first named author was also supported by the National Research Foundation of Korea (NRF) Grant founded by the Korea government (NRF-2013R1A1A2063240), and the second named author by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2009-0093827).
© 2016, Springer Science+Business Media New York.
All Science Journal Classification (ASJC) codes
- Theoretical Computer Science
- Computer Science Applications
- Discrete Mathematics and Combinatorics
- Applied Mathematics