Reducibility of certain class of polynomials over Fp, whose degree depends on p, can be deduced by checking the reducibility of a quadratic and cubic polynomial. This thesis explains how can we speeds up the reducibility procedure.

En considérant les polynômes sur le corps fini de Galois à deux éléments, notre intention porte sur la divisibilité des trinômes x^am+x^bs+1, pour m>s≥1, par un polynôme irréductible de degré r, pour cela, nous avons réalisé le résultat :S'il existe m, s des entiers positifs tels que le trinôme x^am+x^bs+1 soit divisible par un polynôme irréductible de degré r sur F2, alors a et b ne sont pas divisibles par (2r- 1). Pour ce type de trinômes nous conjecturons que le rapport πM(a,b)/ πM(1,1) tend vers une limite finie (dépendant de a et b) quand M tend vers l'infini. Notre recherche porte ensuite sur les codes cycliques de rendement 1/2 sur les deux corps finis F3 et F5 et nous accentuons notre recherche sur ceux iso duaux. Le problème central dans la théorie du codage est trouver la plus grande distance minimum dq pour laquelle un code de paramètres [n, q, d] sur Fq existe. Dans ce contexte nous avons réussi à optimiser cette distance pour les codes cycliques de taux 1/2 sur F3 et F5 en allant jusqu’à la longueur 74 pour les codes ternaires et 42 pour ceux sur F5. Nous avons aussi réussi à construire sept classes de codes cycliques iso-duaux sur le corps fini à 3 éléments et trois classes de codes cycliques iso-duaux sur le corps fini à 5 éléments. / Considering polynomials over the Galois finite fields for two elements, our intention stand over the divisibility of the trinomials x^am+x^bs+1, for m>s ≥ 1, by an irreducible polynomial of degree r, for this, we contribute to the result :If there exist positive integers m, s such that the trinomial x^am+x^bs+1 is divisible by an irreducible polynomial of degree r over F2, then a and b are not divisible by (2^r- 1). For this type of trinomials we conjectured that the ratios πM(a,b)/ πM(1,1) tend to a finite limit (dependently of a and b) when M tend to infinity. Our research stand at sequel on the cyclic codes of rate 1/2 over the two finite fields F3 and F5 and we check our research over whose are isodual. The so-called fundamental problem in coding theory is finding the largest value of dq for which a code of parameters [n, q, d] over Fq exists. In this context we have successfully optimize this distance for the cyclic codes of rate 1/2 over F3 and F5 up to length 74 for the ternary cyclic codes and 42 for whose over F5. We have also successful to construct seven classes of isodual cyclic codes over the field of 3 elements and three classes over the field of 5 elements.

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This dissertation explores an analogue of the prime number theorem for polynomials over finite fields as well as its connection to the necklace factorization algorithm T-transform and the string complexity measure T-complexity. Specifically, a precise asymptotic expansion for the prime polynomial counting function is derived. The approximation given is more accurate than previous results in the literature while requiring very little computational effort. In this context asymptotic series expansions for Lerch transcendent, Eulerian polynomials, truncated polylogarithm, and polylogarithms of negative integer order are also provided. The expansion formulas developed are general and have applications in numerous areas other than the enumeration of prime polynomials. A bijection between the equivalence classes of aperiodic necklaces and monic prime polynomials is utilized to derive an asymptotic bound on the maximal T-complexity value of a string. Furthermore, the statistical behaviour of uniform random sequences that are factored via the T-transform are investigated, and an accurate probabilistic model for short necklace factors is presented. Finally, a T-complexity based conditional string complexity measure is proposed and used to define the normalized T-complexity distance that measures similarity between strings. The T-complexity distance is proven to not be a metric. However, the measure can be computed in linear time and space making it a suitable choice for large data sets. / Graduate / 0544 0984 0405 / nrebenich@gmail.com