Linear Codes over Finite Fields (New) [HB 97]

Changes:

• The function VerfiyMinimumWeight was not performing well. It has been replaced by the function VerifyMinimumWeightLowerBound which determines whether or not a given value is a lower bound of a given code. The function VerifyMinimumWeightUpperBound performs similarly for an upper bound.
• The function SubfieldRepresentation has been replaced by SubfieldRepresentationCode along with the subfield code constructions SubfieldRepresentationParityCode and SubfieldCode.

• The old function ExpurgateCode now allows multiple words to be expurgated, while ExtendCode now allows an extension of arbitrary length, and DirectSum now takes a sequence of arbitrary length. The old function VectorSpace is now also named RSpace.

New Features:

• The advanced Zimmermann algorithm for MinimumWeight has been optimized: it now discards information sets of low rank. A new parameter MaximumTime has been introduced to abort lengthy calculations, and the parameter cutoff has been renamed RankLowerBound.
• A new database gives the user access to the best known binary linear codes of length up to 256 (joint project with Markus Grassl). Codes can be accessed through the functions BestKnownLinearCode, BestDimensionLinearCode, BestLengthLinearCode, and their abbreviations BKLC, BDLC and BLLC respectively. Bounds can be accessed through the functions BKLCLowerBound, BKLCUpperBound, etc. The functions are designed so that any two of the length, dimension, or minimum weight can be specified, and the optimal result for the missing parameter is returned.

• New constructions for codes are CordaroWagnerCode, which returns the dimension 2 Cordaro-Wagner code, MDSCode, which returns a maximum-distance-separable code, QuasiCyclicCode which returns the quasicyclic code formed by concatenating cyclic codes, and NonPrimitveAlternantCode which returns a non-primitive alternant code over $\mbox{\rm GF}(2)$.
• Some new constructions based on circulant matrices and related to quadratic residue codes are DoublyCirculantQRCode and BorderedDoublyCirculantQRCode. The function TwistedQRCode returns a twisted quadratic residue code. A generalisation PowerResidueCode returns the code of an arbitrary power residue.
• New functions which take a code and produce a new code are: ResidueCode which performs a single Griesmer step, PadCode which extends each codeword with zeros, ConstructionY1 which performs construction Y1, and ExpurgateWeightCode which expurgates all codewords of given weight.
• New functions which take multiple codes to produce new codes are: CodeComplement which returns the complement subspace from a given subcode, CodeJuxtaposition which joins the generator matrices of two codes, `cat' which concatenates all combinations of codewords from two codes, and ZinovievCode which creates generalised concatenated codes. Further such functions include ConstructionX, ConstructionX3 and ConstructionXX.
• New functions relating to subcodes are: Subcode which returns a subcode of given dimension, and SubcodeBetweenCode which returns a code nested between an existing code-subcode pair.

• The new function SubcodeWordsOfWeight returns the subcode generated by codewords of specified weights.
• The new function WordsOfBoundedWeight returns all code words whose length lies between the given bounds.

• Machinery has been provided which allows the construction of algebraic-geometric codes. See functions AlgebraicGeometricCode and AGCode.