Algebraic-Geometric NRT-Codes with Known Gap Numbers

In [1] algebraic-geometric codes are generalized to NRT-space: Let F denote a global function field with full constant field Fb and genus g = g(F/Fb). Furthermore let P1,…, Ps denote s rational places of F and let G denote a divisor of F with suppG∩{P1,…, Ps} = ∅ and deg G < Ts. The algebraic-geometric NRT-code AG(T ;F, G) is defined as

AG(T ;F, G) := {(f(T −1)(Pi),…, f(0)(Pi))i=1,…, s  :  fL(G)},

where L(G) denotes the Riemann-Roch space of the divisor G and f(j)(P) denotes the jth coefficient in the Taylor expansion of f at P with respect to some local parameter defined over Fb. It is a linear [(s, T ), n, d]-NRT-code over Fb with n = dim G and dsT – deg G.

The Dimension of Algebraic-Geometric NRT-Codes

It follows from the definition of the genus that dim G ≥ deg Gg + 1, i.e., the code contains a subcode with dimension n ≥ deg Gg + 1. If deg G ≥ 2g, this bound is sharp. For deg G < 2g it is possible that dim G > deg Gg + 1. If G is of the form rP, with P denoting a place of F, the exact value of dim G = dim rP is determined by the Weierstrass semigroup of P. For some function fields these numbers are known and MinT uses this information.

In addition to that, the dimension of the zero-divisor is always 1 because L(0) is the space of constant functions. Therefore AG(F, 0) is the [(s, T ), 1, sT ]-NRT-code {(0,…, 0, a)i=1,…s  :  aFb} for all function fields F.

Construction of the Divisor G

If the number of rational places N(F) is larger than s, one can choose one of these places P and use G = rP. Obviously, deg G = deg rP = r. Thus AG(T ;F, rP) is an [(s, T ), max{1, rg + 1}, sT r]-code with s = N(F) − 1 for all r = 0,…, sT −1.

If a code of length s = N(F) is to be obtained, places of larger degree have to be used for the construction of G. MinT uses divisors G of the form G = Q + rP, where P is some non-rational place and Q is the sum of zero or more other non-rational places.

Subcode Structure

If GGʹ, then AG(T ;F, G) is a subcode of AG(T ;F, Gʹ) and construction X can be applied to this pair of codes. In particular this situation arises for AG(T ;F, Q + rP) ⊆ AG(T ;F, Q + rʹP) with Q denoting an arbitrary divisor (but usually the zero-divisor), P denoting a place, and integers r, rʹ with r < rʹ.

If P is one of the rational places, we have AG(T ;F,(rT + 1)P) ⊆ ⋯ ⊆ AG(T ;F, rP) and standard lengthening for NRT-codes can be applied. The result is an [(s, T ), max{1, rg + 1}, sT r]-code with s = N(F), namely the so-called extended algebraic-geometric NRT-code AGe(T ;F, rP).

Special Cases

If F is the rational function field over Fb, P1,…, Pb are the b places corresponding to the zeros of the linear functions, and G = rP, Reed-Solomon NRT-codes RS(T , r + 1, b) are obtained.

See Also

References

[1]Michael Yu. Rosenbloom and Michael A. Tsfasman.
Codes for the m-metric.
Problems of Information Transmission, 33:55–63, 1997.

Copyright

Copyright © 2004, 2005, 2006, 2007, 2008, 2009, 2010 by Rudolf Schürer and Wolfgang Ch. Schmid.
Cite this as: Rudolf Schürer and Wolfgang Ch. Schmid. “Algebraic-Geometric NRT-Codes with Known Gap Numbers.” From MinT—the database of optimal net, code, OA, and OOA parameters. Version: 2008-04-04. http://mint.sbg.ac.at/desc_OGoppa-gapnumbers.html

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