## Extended Algebraic-Geometric NRT-Codes

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

*T*;

*F*,

*G*) := {(

*f*

^{(T −1)}(

*P*

_{i}),…,

*f*

^{(0)}(

*P*

_{i}))

_{i=1,…, s}:

*f*∈ L(

*G*)},

where L(*G*) denotes the Riemann-Roch space of the divisor *G* and *f*^{(j)}(*P*) denotes the *j*th coefficient in the Taylor expansion of *f* at *P* with respect to some local parameter defined over **F**_{b}. It is a linear [(*s*, *T *), *n*, *d*]-NRT-code over **F**_{b} with *n* = dim *G* and *d* ≥ *sT * – deg *G*.

### The Dimension of Algebraic-Geometric NRT-Codes

It follows from the definition of the genus that dim *G* ≥ deg *G*−*g* + 1, i.e., the code contains a subcode with dimension *n* ≥ deg *G*−*g* + 1. If deg *G* ≥ 2*g*, this bound is sharp. For deg *G* < 2*g* it is possible that dim *G* > deg *G*−*g* + 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} : *a* ∈ **F**_{b}} 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, *r* – *g* + 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 *G* ≤ *G*ʹ, then AG(*T *;*F*, *G*) is a subcode of AG(*T *;*F*, *G*ʹ) 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*, *r*ʹ with *r* < *r*ʹ.

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

### Special Cases

If *F* is the rational function field over **F**_{b}, *P*_{1},…, *P*_{b} 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

For

*T*= 1 the normal algebraic-geometric codes are obtained.

### 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. “Extended Algebraic-Geometric NRT-Codes.”
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Version: 2015-09-03.
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