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There is a local ring I of order 4, without identity for the multiplication, defined by generators and relations as I = ⟨ a , b ∣ 2 a = 2 b = 0 , a 2 = b , a b = 0 ⟩. We study a recursive construction of self-orthogonal codes over I. We classify self orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 6. In particular, we classify Type IV codes (QSD codes with even weights) and quasi Type IV codes (QSD codes with even torsion code) up to n = 6. [ABSTRACT FROM AUTHOR]

The original algorithm of D.V. Chudnovsky and G.V. Chudnovsky for the multiplication in extensions of finite fields provides a bilinear complexity which is uniformly linear with respect to the degree of the extension. Recently, Randriambololona generalized the method, allowing asymmetry in the interpolation procedure. The aim of this article is to make effective this method. We first make explicit this generalization in order to construct the underlying asymmetric algorithms. Then, we propose a generic strategy to construct these algorithms using places of higher degrees and without derivated evaluation. Finally, we provide examples of three multiplication algorithms along with their Magma implementation: in F 16 13 using only rational places, in F 4 5 using also places of degree two, and in F 2 5 using also places of degree four. [ABSTRACT FROM AUTHOR]

There is a local ring E of order 4 , without identity for the multiplication, defined by generators and relations as E = 〈 a , b | 2 a = 2 b = 0 , a 2 = a , b 2 = b , a b = a , b a = b 〉. We study a recursive construction of self-orthogonal codes over E. We classify, up to permutation equivalence, self-orthogonal codes of length n and size 2 n (called here quasi self-dual codes or QSD) up to the length n = 1 2. In particular, we classify Type IV codes (QSD codes with even weights) up to n = 1 2. [ABSTRACT FROM AUTHOR]

Thanks to a new construction of the so-called Chudnovsky- Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation and they allow computations to be parallelized while maintaining a low number of bilinear multiplications. We give an example with the finite field 픽 1613. [ABSTRACT FROM AUTHOR]

We indicate a strategy in order to construct bilinear multiplication algorithms of type Chudnovsky in large extensions of any finite field. In particular, using the symmetric version of the generalization of Randriambololona specialized on the elliptic curves, we show that it is possible to construct such algorithms with low bilinear complexity. More precisely, if we only consider the Chudnovsky-type algorithms of type symmetric elliptic, we show that the symmetric bilinear complexity of these algorithms is in where n corresponds to the extension degree, and is the iterated logarithm. Moreover, we show that the construction of such algorithms can be done in time polynomial in n. Finally, applying this method we present the effective construction, step by step, of such an algorithm of multiplication in the finite field 픽357. [ABSTRACT FROM AUTHOR]

The Chudnovsky algorithm for the multiplication in extensions of finite fields provides a bilinear complexity uniformly linear with respect to the degree of the extension. Recently, Randriambololona has generalized the method, allowing asymmetry in the interpolation procedure and leading to new upper bounds on the bilinear complexity. In this note, we describe the construction of this asymmetric method without derived evaluation. To do this, we translate this generalization into the language of algebraic function fields and we give a strategy of construction and implementation. [ABSTRACT FROM AUTHOR]

Thanks to a new construction of the Chudnovsky and Chudnovsky multiplication algorithm, we design efficient algorithms for both the exponentiation and the multiplication in finite fields. They are tailored to hardware implementation and they allow computations to be parallelized, while maintaining a low number of bilinear multiplications. [ABSTRACT FROM AUTHOR]

There exist two semi-local rings of order 6 without identity for the multiplication. We classify up to coordinate permutation self-orthogonal codes of length n and size 6n/2 over these rings (called here quasi self-dual codes or QSD) till the length n=8. To any such code is attached canonically a Ze-code, which, when self-dual, produces an unimodular lattice by Construction A. [ABSTRACT FROM AUTHOR]