On Geometric Goppa Codes from Elementary Abelian p -Extensions of $${{\mathbb{F}}}_{{p}^{s}}(x)$$ F p s (
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DING THEORY
On Geometric Goppa Codes from Elementary Abelian p-Extensions of Fps (x) N. Patankera, ∗ and S. K. Singha, ∗∗ a
Indian Institute of Science Education and Research, Bhopal, India e-mail : ∗ [email protected], ∗∗ [email protected]
Received February 12, 2020; revised June 15, 2020; accepted June 30, 2020 Abstract—Let p be a prime number and s > 0 an integer. In this short note, we investigate one-point geometric Goppa codes associated with an elementary abelian p-extension of Fps (x). We determine their dimension and exact minimum distance in a few cases. These codes are a special case of weak Castle codes. We also list exact values of the second generalized Hamming weight of these codes in a few cases. Simple criteria for self-duality and quasi-self-duality of these codes are also provided. Furthermore, we construct examples of quantum codes, convolutional codes, and locally recoverable codes on the function field. Key words: elementary abelian p-extension of Fps (x), geometric Goppa codes, generalized Hamming weight. DOI: 10.1134/S0032946020030035
1. INTRODUCTION Let Fps be the finite field with ps elements of characteristic p (where s is a positive integer). A linear code is a Fps -subspace of Fnps , the n-dimensional standard vector space over Fps . Such codes can be used for transmission of information. It was observed by Goppa in [1] that we can use divisors in a field of algebraic functions to construct a class of linear codes. In Goppa’s construction, we choose a divisor G and n rational places (i.e., places of degree one) of the algebraic function field to form a linear code of length n. These codes are called geometric Goppa codes. If G is of the form rQ, for a place Q of the algebraic function field and integer r, then these codes are called one-point codes. One-point geometric Goppa codes over algebraic function field of Hermitian curves have been studied in [2–5], etc. Elementary abelian p-extension of the rational function field Fps (x) is a Galois extension F of Fps (x) such that Gal(F/Fps (x)) is an elementary abelian group of exponent p. The algebraic function field associated with a Hermitian curve is an example of such an extension. The properties of elementary abelian p-extensions of Fps (x) have been studied in [2, 6–8], etc. Another example of elementary abelian p-extension of Fps (x) is the function field Fps (x, y)/Fps defined by A(y) = B(x), where A(T ) ∈ Fps [T ] is a separable additive polynomial of degree q = pt , for some t, such that all its roots are contained in Fps and the degree of B(T ) ∈ Fps [T ] is not divisible by p. The nonsingular projective curve X associated with this function field was studied in [8]. In [8] the authors determined parameters of geometric Goppa codes CL (D, G) on this function field with the assumption that deg D ≥ 4g − 2. In this note we study one-point geometric Goppa codes on an elementary abelian p-extension of Fps (x) without the above assumption. A special type of elementary abelian p-extensions of Fps (x) considered in the present note are exampl
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