It is well-known that any property of Reed–Muller codes is a property of Boolean functions. Reed Muller codes provide a natural way to quantify the degree, the nonlinearity, the correlation-immunity or the propagation characteristics of a Boolean function [1]. On the other hand, Reed Muller codes are an important class of error-correcting codes, in particular they can be viewed as extended cyclic codes. They play a crucial role in the study of important families of cryptographic mappings, such as permutations on finite fields.
Here, we present the multivariable definition of Reed-Muller codes. More on Reed Muller codes can be found in [2].
Definition 1.Define\(\{\mathbf{F}_2^m,+\}\)as an ordered vector space:
where\({\bf v}_i\)is an m-dimensional binary vector (often onetakes\({\bf v}_i\)as the binary representation of integer i). The Reed-Muller code of length 2mand order r,\(0\leq r\leq m\), denoted by\({\cal...
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References
Massey, J.L. (1995). “Some applications of coding theory in cryptography.” Codes and Ciphers: Cryptography and Coding IV, ed. P.G. Farell. Formara Ltd., Essex, England, 33–47.
Pless, V.S., W.C. Huffman, and R.A. Brualdi (1998). “An introduction to algebraic codes.” Handbook of Coding Theory, Part 1: Algebraic Coding, Chapter 1. Elsevier, Amsterdam, The Netherlands.
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Charpin, P. (2005). Reed–Muller Codes. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_348
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DOI: https://doi.org/10.1007/0-387-23483-7_348
Publisher Name: Springer, Boston, MA
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