We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds, which are an improvement of a bound by Zinoviev, Litsyn and Laihonen, are independent of other classical known theoretical bounds. Among these, we mention the Griesmer bound for linear codes, of which we provide a partial generalization for the systematic case. Our experiments show that in some cases (the majority of cases for some q) our bounds provide the best value, compared to all other closed-formula upper-bounds. We also present an algebraic method for computing the minimum weight of any nonlinear code. We show that for some particular code, using a non-standard representation of the code, our method is faster than brute force. An application of this method allows to compute the nonlinearity of a Boolean function, improving existing algebraic methods and reaching the same complexity of algorithms based on the fast Fourier transform.

Computational techniques for nonlinear codes and Boolean functions / Bellini, Emanuele. - (2014), pp. 1-182.

Computational techniques for nonlinear codes and Boolean functions

Bellini, Emanuele
2014-01-01

Abstract

We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds, which are an improvement of a bound by Zinoviev, Litsyn and Laihonen, are independent of other classical known theoretical bounds. Among these, we mention the Griesmer bound for linear codes, of which we provide a partial generalization for the systematic case. Our experiments show that in some cases (the majority of cases for some q) our bounds provide the best value, compared to all other closed-formula upper-bounds. We also present an algebraic method for computing the minimum weight of any nonlinear code. We show that for some particular code, using a non-standard representation of the code, our method is faster than brute force. An application of this method allows to compute the nonlinearity of a Boolean function, improving existing algebraic methods and reaching the same complexity of algorithms based on the fast Fourier transform.
2014
XXVI
2013-2014
Matematica (29/10/12-)
Mathematics
Sala, Massimiliano
no
Inglese
Settore INF/01 - Informatica
Settore MAT/02 - Algebra
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/369066
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