# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022047
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## Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators

 Universities of Bergen, Norway, and Paris 8, France

Received  March 2022 Revised  May 2022 Early access June 2022

Fund Project: The research of the author is partly supported by the Trond Mohn Foundation and Norwegian Research Council

We consider the algebraic normal form (ANF) of the indicators (i.e. characteristic functions) of linear binary codes, and characterize the minimum distance of such codes in a very simple way by means of this ANF. We extend this characterization to nonlinear binary codes, via another representation, the numerical normal form (NNF). We further extend these characterizations to linear codes over finite fields and (after introducing a generalization of the NNF to functions from ${\Bbb F}_p^n$ to ${\Bbb R}$) to unrestricted codes over prime fields. We also study the weight distribution by means of the NNF, and the covering radius of binary codes with the same approach; the latter is more difficult to address, but we obtain some results as well.

Citation: Claude Carlet. Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators. Advances in Mathematics of Communications, doi: 10.3934/amc.2022047
##### References:
 [1] P. Beelen and G. Leander, A new construction of highly nonlinear S-boxes, Cryptogr. Commun., 4 (2012), 65-77.  doi: 10.1007/s12095-011-0052-4. [2] C. Carlet, Handling vectorial functions by means of their graph indicators, IEEE Trans. Inform. Theory, 66 (2020), 6324-6339.  doi: 10.1109/TIT.2020.2981524. [3] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Monograph Cambridge University Press, 2021. [4] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [5] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [6] P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Information and Control, 23 (1973), 407-438. [7] C. Ding, Cyclic Codes from some monomials and trinomials, SIAM J. Discrete Math., 27 (2013), 1977-1994.  doi: 10.1137/120882275. [8] C. Ding, A construction of binary linear codes from boolean functions, Discrete Math., 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029. [9] C. Ding, A sequence construction of cyclic codes over finite fields, Cryptogr. Commun., 10 (2018), 319-341.  doi: 10.1007/s12095-017-0222-0. [10] C. Ding, A. Munemasa and V. D. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inform. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401. [11] C. Ding and Z. Zhou, Binary cyclic codes from explicit polynomials over $GF(2^m)$, Discrete Math., 321 (2014), 76-89.  doi: 10.1016/j.disc.2013.12.020. [12] A. M. Kerdock, A class of low-rate non linear codes, Information and Control, 20 (1972), 182-187. [13] D. Kleitman, On Dedekind's problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc., 21 (1969), 677-682.  doi: 10.2307/2036446. [14] J. Liu, S. Mesnager and L. Chen, On the nonlinearity of S-boxes and linear codes, Cryptogr. Commun., 9 (2017), 345-361.  doi: 10.1007/s12095-015-0176-z. [15] F. J. MacWilliams and N. J. Sloane, The Theory of Error-correcting Codes, Amsterdam-New York-Oxford, 1977. [16] S. Mesnager, Bent vectorial functions and linear codes from o-polynomials, Des. Codes Cryptogr., 77 (2015), 99-116.  doi: 10.1007/s10623-014-9989-6. [17] S. Mesnager, "Linear codes from functions", Concise Encyclopedia Coding Theory, CRC Press/Taylor and Francis Group (Publisher), London, New York, 20 (2021), 94 pp. [18] D. Popescu, The algebraic degree of perfect binary codes, IEEE Trans. Inform. Theory, 54 (2008), 5198-5202.  doi: 10.1109/TIT.2008.929971. [19] D. Tang, C. Carlet and Z. Zhou, Binary linear codes from vectorial boolean functions and their weight distribution, Discrete Math., 340 (2017), 3055-3072.  doi: 10.1016/j.disc.2017.07.008. [20] J. Wolfmann, Bent functions and coding theory, Difference Sets, Sequences and Their Correlation Properties, 542 (1999), 393-4189. [21] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Transactions on Information Theory, 52 (2006), 712-717.  doi: 10.1109/TIT.2005.862125.

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##### References:
 [1] P. Beelen and G. Leander, A new construction of highly nonlinear S-boxes, Cryptogr. Commun., 4 (2012), 65-77.  doi: 10.1007/s12095-011-0052-4. [2] C. Carlet, Handling vectorial functions by means of their graph indicators, IEEE Trans. Inform. Theory, 66 (2020), 6324-6339.  doi: 10.1109/TIT.2020.2981524. [3] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Monograph Cambridge University Press, 2021. [4] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [5] C. Carlet, C. Ding and J. Yuan, Linear codes from perfect nonlinear mappings and their secret sharing schemes, IEEE Trans. Inform. Theory, 51 (2005), 2089-2102.  doi: 10.1109/TIT.2005.847722. [6] P. Delsarte, Four fundamental parameters of a code and their combinatorial significance, Information and Control, 23 (1973), 407-438. [7] C. Ding, Cyclic Codes from some monomials and trinomials, SIAM J. Discrete Math., 27 (2013), 1977-1994.  doi: 10.1137/120882275. [8] C. Ding, A construction of binary linear codes from boolean functions, Discrete Math., 339 (2016), 2288-2303.  doi: 10.1016/j.disc.2016.03.029. [9] C. Ding, A sequence construction of cyclic codes over finite fields, Cryptogr. Commun., 10 (2018), 319-341.  doi: 10.1007/s12095-017-0222-0. [10] C. Ding, A. Munemasa and V. D. Tonchev, Bent vectorial functions, codes and designs, IEEE Trans. Inform. Theory, 65 (2019), 7533-7541.  doi: 10.1109/TIT.2019.2922401. [11] C. Ding and Z. Zhou, Binary cyclic codes from explicit polynomials over $GF(2^m)$, Discrete Math., 321 (2014), 76-89.  doi: 10.1016/j.disc.2013.12.020. [12] A. M. Kerdock, A class of low-rate non linear codes, Information and Control, 20 (1972), 182-187. [13] D. Kleitman, On Dedekind's problem: The number of monotone Boolean functions, Proc. Amer. Math. Soc., 21 (1969), 677-682.  doi: 10.2307/2036446. [14] J. Liu, S. Mesnager and L. Chen, On the nonlinearity of S-boxes and linear codes, Cryptogr. Commun., 9 (2017), 345-361.  doi: 10.1007/s12095-015-0176-z. [15] F. J. MacWilliams and N. J. Sloane, The Theory of Error-correcting Codes, Amsterdam-New York-Oxford, 1977. [16] S. Mesnager, Bent vectorial functions and linear codes from o-polynomials, Des. Codes Cryptogr., 77 (2015), 99-116.  doi: 10.1007/s10623-014-9989-6. [17] S. Mesnager, "Linear codes from functions", Concise Encyclopedia Coding Theory, CRC Press/Taylor and Francis Group (Publisher), London, New York, 20 (2021), 94 pp. [18] D. Popescu, The algebraic degree of perfect binary codes, IEEE Trans. Inform. Theory, 54 (2008), 5198-5202.  doi: 10.1109/TIT.2008.929971. [19] D. Tang, C. Carlet and Z. Zhou, Binary linear codes from vectorial boolean functions and their weight distribution, Discrete Math., 340 (2017), 3055-3072.  doi: 10.1016/j.disc.2017.07.008. [20] J. Wolfmann, Bent functions and coding theory, Difference Sets, Sequences and Their Correlation Properties, 542 (1999), 393-4189. [21] J. Yuan, C. Carlet and C. Ding, The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Transactions on Information Theory, 52 (2006), 712-717.  doi: 10.1109/TIT.2005.862125.
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