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In this paper, we focus on extensions of methods for interpolating rational functions from their evaluations, in the context of erroneous evaluations. This problem can be seen both from a computer algebra and a coding theory point of view. In computer algebra, this is a generalization of Simultaneous Rational Function Reconstruction with errors and multiprecision evaluations. From an error correcting codes point of view, this problem is related to the decoding of some algebraic codes such as Reed-Solomon, Derivatives or Multiplicity codes. We give conditions on the inputs of the problem which guarantee the uniqueness of the interpolant. Since we deal with rational functions, some evaluation points may be poles: a first contribution of this work is to correct any error in a scenario with poles and multiplicities that extends (Kaltofen et al., 2020). Our second contribution is to adapt rational function reconstruction for random errors , and provide better conditions for uniqueness using interleaving techniques as in (Guerrini et al., 2021). [ABSTRACT FROM AUTHOR]

A set of congruence relations is a \mathbb Z -covering if each integer belongs to at least one congruence class from that set. In this paper, we first show that most existing scalar multiplication algorithms can be formulated in terms of covering systems of congruences. Then, using a special form of covering systems called exact $n$ -covers, we present a novel uniformly randomized scalar multiplication algorithm with built-in protections against most passive side-channel attacks. Our algorithm randomizes the addition chain using a mixed-radix representation of the scalar. Its reduced overhead and purposeful robustness could make it a sound replacement to several conventional countermeasures. In particular, it is significantly faster than Coron's scalar blinding technique for elliptic curves when the choice of a particular finite field tailored for speed compels to double the size of the scalar, hence the cost of the scalar multiplication. [ABSTRACT FROM AUTHOR]

Most bounds on the size of codes hold for any code, whether linear or not. Notably, the Griesmer bound holds only in the linear case and so optimal linear codes are not necessarily optimal codes. In this paper, we identify code parameters $(q,d,k)$ , namely, field size, minimum distance, and combinatorial dimension, for which the Griesmer bound also holds in the (systematic) nonlinear case. Moreover, we show that the Griesmer bound does not necessarily hold for a systematic code by explicit construction of a family of optimal systematic binary codes. On the other hand, we are able to provide some versions of the Griesmer bound holding for all the systematic codes. [ABSTRACT FROM AUTHOR]

We present some upper bounds on the size of nonlinear codes and their restriction to systematic codes and linear codes. These bounds are independent of other known theoretical bounds, e.g., the Griesmer bound, the Johnson bound, the Plotkin bound, and of linear programming bounds. One of the new bound is actually an improvement of a bound by Zinoviev, Litsyn, and Laihonen. Our experiments show that in the linear case our bounds provide the best value in a wide range, compared with all other closed-formula upper bounds. In the nonlinear case, we also compare our bound with the linear programming bound and with some improvements on it, show that there are cases where we beat these bounds. In particular, we obtain a new bound in Brouwer's table for A3(16,3). [ABSTRACT FROM AUTHOR]

Abstract: An identifying code of a graph is a dominating set such that every vertex of is distinguished from other vertices by the set of vertices in that are at distance at most 1 from . The problem of finding an identifying code of minimum possible size turned out to be a challenging problem. It was proved by N. Bertrand, I. Charon, O. Hudry and A. Lobstein that if a graph on vertices with at least one edge admits an identifying code, then a minimal identifying code has size at most . They introduced classes of graphs whose smallest identifying code is of size . Few conjectures were formulated to classify the class of all graphs whose minimum identifying code is of size . In this paper, disproving these conjectures, we classify all finite graphs for which all but one of the vertices are needed to form an identifying code. We classify all infinite graphs needing the whole set of vertices in any identifying code. New upper bounds in terms of the number of vertices and the maximum degree of a graph are also provided. [Copyright &y& Elsevier]

The most important families of nonlinear codes are systematic. A brute-force check is the only known method to compute their weight distribution and distance distribution. On the other hand, it outputs also all closest word pairs in the code. In the black-box complexity model, the check is optimal among closest-pair algorithms. In this paper, we provide a Gröbner basis technique to compute the weight/distance distribution of any systematic nonlinear code. Also our technique outputs all closest pairs. Unlike the check, our method can be extended to work on code families. [ABSTRACT FROM AUTHOR]