Duals of Affine Grassmann Codes and Their Relatives

P. Beelen; S. R. Ghorpade; T. Hoholdt

doi:10.1109/TIT.2012.2187171

Duals of Affine Grassmann Codes and Their Relatives

P. Beelen, S. R. Ghorpade, T. Hoholdt

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Abstract

Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.

Original language	English
Journal	I E E E Transactions on Information Theory
Volume	58
Issue number	6
Pages (from-to)	3843-3855
ISSN	0018-9448
DOIs	https://doi.org/10.1109/TIT.2012.2187171
Publication status	Published - 2012

Keywords

Linear codes
Reed-Muller codes
Electronic mail
Frequency modulation
Linear code
Parity check codes
Polynomials
Sparse matrices

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10.1109/TIT.2012.2187171

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title = "Duals of Affine Grassmann Codes and Their Relatives",

abstract = "Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.",

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T1 - Duals of Affine Grassmann Codes and Their Relatives

AU - Beelen, P.

AU - Ghorpade, S. R.

AU - Hoholdt, T.

PY - 2012

Y1 - 2012

N2 - Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.

AB - Affine Grassmann codes are a variant of generalized Reed-Muller codes and are closely related to Grassmann codes. These codes were introduced in a recent work by Beelen Here, we consider, more generally, affine Grassmann codes of a given level. We explicitly determine the dual of an affine Grassmann code of any level and compute its minimum distance. Further, we ameliorate the results by Beelen concerning the automorphism group of affine Grassmann codes. Finally, we prove that affine Grassmann codes and their duals have the property that they are linear codes generated by their minimum-weight codewords. This provides a clean analogue of a corresponding result for generalized Reed-Muller codes.

KW - Linear codes

KW - Reed-Muller codes

KW - Electronic mail

KW - Frequency modulation

KW - Linear code

KW - Parity check codes

KW - Polynomials

KW - Sparse matrices

U2 - 10.1109/TIT.2012.2187171

DO - 10.1109/TIT.2012.2187171

M3 - Journal article

SN - 0018-9448

VL - 58

SP - 3843

EP - 3855

JO - I E E E Transactions on Information Theory

JF - I E E E Transactions on Information Theory

IS - 6

ER -

Duals of Affine Grassmann Codes and Their Relatives

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