20,298 research outputs found

    p-Adic estimates of Hamming weights in Abelian codes over Galois rings

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    A generalization of McEliece's theorem on the p-adic valuation of Hamming weights of words in cyclic codes is proved in this paper by means of counting polynomial techniques introduced by Wilson along with a technique known as trace-averaging introduced here. The original theorem of McEliece concerned cyclic codes over prime fields. Delsarte and McEliece later extended this to Abelian codes over finite fields. Calderbank, Li, and Poonen extended McEliece's original theorem to cover cyclic codes over the rings /spl Zopf//sub 2//sup d/, Wilson strengthened their results and extended them to cyclic codes over /spl Zopf//sub p//sup d/, and Katz strengthened Wilson's results and extended them to Abelian codes over /spl Zopf//sub p//sup d/. It is natural to ask whether there is a single analogue of McEliece's theorem which correctly captures the behavior of codes over all finite fields and all rings of integers modulo prime powers. In this paper, this question is answered affirmatively: a single theorem for Abelian codes over Galois rings is presented. This theorem contains all previously mentioned results and more

    Multivariate Ap\'ery numbers and supercongruences of rational functions

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    One of the many remarkable properties of the Ap\'ery numbers A(n)A (n), introduced in Ap\'ery's proof of the irrationality of ζ(3)\zeta (3), is that they satisfy the two-term supercongruences \begin{equation*} A (p^r m) \equiv A (p^{r - 1} m) \pmod{p^{3 r}} \end{equation*} for primes p5p \geq 5. Similar congruences are conjectured to hold for all Ap\'ery-like sequences. We provide a fresh perspective on the supercongruences satisfied by the Ap\'ery numbers by showing that they extend to all Taylor coefficients A(n1,n2,n3,n4)A (n_1, n_2, n_3, n_4) of the rational function \begin{equation*} \frac{1}{(1 - x_1 - x_2) (1 - x_3 - x_4) - x_1 x_2 x_3 x_4} . \end{equation*} The Ap\'ery numbers are the diagonal coefficients of this function, which is simpler than previously known rational functions with this property. Our main result offers analogous results for an infinite family of sequences, indexed by partitions λ\lambda, which also includes the Franel and Yang--Zudilin numbers as well as the Ap\'ery numbers corresponding to ζ(2)\zeta (2). Using the example of the Almkvist--Zudilin numbers, we further indicate evidence of multivariate supercongruences for other Ap\'ery-like sequences.Comment: 19 page

    Lacunary formal power series and the Stern-Brocot sequence

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    Let F(X)=n0(1)εnXλnF(X) = \sum_{n \geq 0} (-1)^{\varepsilon_n} X^{-\lambda_n} be a real lacunary formal power series, where εn=0,1\varepsilon_n = 0, 1 and λn+1/λn>2\lambda_{n+1}/\lambda_n > 2. It is known that the denominators Qn(X)Q_n(X) of the convergents of its continued fraction expansion are polynomials with coefficients 0,±10, \pm 1, and that the number of nonzero terms in Qn(X)Q_n(X) is the nnth term of the Stern-Brocot sequence. We show that replacing the index nn by any 2-adic integer ω\omega makes sense. We prove that Qω(X)Q_{\omega}(X) is a polynomial if and only if ωZ\omega \in {\mathbb Z}. In all the other cases Qω(X)Q_{\omega}(X) is an infinite formal power series, the algebraic properties of which we discuss in the special case λn=2n+11\lambda_n = 2^{n+1} - 1.Comment: to appear in Acta Arithmetic
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