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Lower Bounds on the State Complexity of Geometric Goppa Codes
Authors:T Blackmore  G H Norton
Institution:(1) Infineon Technologies, Stoke Gifford, BS34 8HP, UK;(2) Dept. Mathematics, University of Queensland, Brisbane, 4072
Abstract:We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg 
$$G \leqslant \frac{{n - 2}}{2}{\text{ or deg }}G \geqslant \frac{{n - 2}}{2} + 2g$$
then the state complexity of 
$$C_\mathcal{L} (D,G)$$
is equal to the Wolf bound. For deg 
$$G \in \frac{{n - 1}}{2},\frac{{n - 3}}{2} + 2g]$$
, we use Clifford's theorem to give a simple lower bound on the state complexity of 
$$C_\mathcal{L} (D,G)$$
. We then derive two further lower bounds on the state space dimensions of 
$$C_\mathcal{L} (D,G)$$
in terms of the gonality sequence of 
$$F/\mathbb{F}_q $$
. (The gonality sequence is known for many of the function fields of interest for defining geometric Goppa codes.) One of the gonality bounds uses previous results on the generalised weight hierarchy of 
$$C_\mathcal{L} (D,G)$$
and one follows in a straightforward way from first principles; often they are equal. For Hermitian codes both gonality bounds are equal to the DLP lower bound on state space dimensions. We conclude by using these results to calculate the DLP lower bound on state complexity for Hermitian codes.
Keywords:geometric Goppa codes  Hermitian codes  state complexity  gonality sequence  dimension/length profiles  Clifford's theorem
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