Subquadratic Time Encodable Codes Beating the Gilbert–Varshamov Bound

Narayanan, Anand Kumar; Weidner, Matthew

doi:10.1109/tit.2019.2930538

Cited by 3 publications

(2 citation statements)

References 47 publications

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“…In [30] an encoding algorithm is given which is faster than Opn 3{2 q for certain carefully tailored, asymptotically good sub-codes of AG codes arising from the Garcia-Stichtenoth tower [10]. Our methods do not handle these codes, so the results can not be directly compared.…”

Section: Related Work On Encodingmentioning

confidence: 99%

Fast Encoding of AG Codes Over C_abCurves

Beelen

Rosenkilde

Solomatov

2021

IEEE Trans. Inform. Theory

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We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called C ab curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some C ab curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity Õpn 3{2 q resp. Õpqnq for AG codes over any C ab curve satisfying very mild assumptions, where n is the code length and q the base field size, and Õ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, for example the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity Õpnq for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding and unencoding algorithms have complexities Õpn 5{4 q and Õpn 3{2 q respectively.

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Section: Related Work On Encodingmentioning

confidence: 99%

Fast Encoding of AG Codes Over C_abCurves

Beelen

Rosenkilde

Solomatov

2021

IEEE Trans. Inform. Theory

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“…For the particularly simple case of RS codes, it is classical that they can be encoded in quasi-linear complexity [16] by univariate multipoint evaluation (see Section II-C). For carefully tailored asymptotically good sub-codes of AG codes arising from the Garcia-Stichtenoth tower [10], [27] give a encoding algorithm which is faster than Opn 3{2 q by reducing the encoding to matrix multiplication.…”

mentioning

confidence: 99%

Fast Encoding of AG Codes over $C_{ab}$ Curves

Beelen¹,

Rosenkilde²,

Solomatov³

2020

Preprint

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We investigate algorithms for encoding of one-point algebraic geometry (AG) codes over certain plane curves called C ab curves, as well as algorithms for inverting the encoding map, which we call "unencoding". Some C ab curves have many points or are even maximal, e.g. the Hermitian curve. Our encoding resp. unencoding algorithms have complexity Õpn 3{2 q resp. Õpqnq for AG codes over any C ab curve satisfying very mild assumptions, where n is the code length and q the base field size, and Õ ignores constants and logarithmic factors in the estimate. For codes over curves whose evaluation points lie on a grid-like structure, notably the Hermitian curve and norm-trace curves, we show that our algorithms have quasi-linear time complexity Õpnq for both operations. For infinite families of curves whose number of points is a constant factor away from the Hasse-Weil bound, our encoding algorithm has complexity Õpn 5{4 q while unencoding has Õpn 3{2 q.

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