# Lower Bounds on the State Complexity of Geometric Goppa Codes

Blackmore, T. and Norton, G. H. (2002) Lower Bounds on the State Complexity of Geometric Goppa Codes. Designs Codes and Cryptography, 25 1: 95-115. doi:10.1023/A:1012512718264

Author Blackmore, T.Norton, G. H. Lower Bounds on the State Complexity of Geometric Goppa Codes Designs Codes and Cryptography   Check publisher's open access policy 0925-1022 2002-01 Article (original research) 10.1023/A:1012512718264 25 1 95 115 21 D. JungnickelJ. D. KeyS. A. Vanstone United States Kluwer Academic Press 2002 eng C1230103 Rings And Algebras780101 Mathematical sciences01 Mathematical Sciences We reinterpret the state space dimension equations for geometric Goppa codes. An easy consequence is that if deg G less than or equal to n-2/2 or deg G greater than or equal to n-2/2 + 2g then the state complexity of C-L(D, G) is equal to the Wolf bound. For deg G is an element of [n-1/2, n-3/2 + 2g], we use Clifford's theorem to give a simple lower bound on the state complexity of C-L(D, G). We then derive two further lower bounds on the state space dimensions of C-L(D, G) in terms of the gonality sequence of F/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-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. Computer Science, Theory & MethodsMathematics, AppliedGeometric Goppa CodesHermitian CodesState ComplexityGonality SequenceDimension/length ProfilesClifford's TheoremGeneralized Hamming WeightsHierarchy C1

 Document type: Journal Article Article (original research) Excellence in Research Australia (ERA) - Collection School of Physical Sciences Publications

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