On a family of Abelian codes and their state complexities

Blackmore, Tim; Norton, Graham H.

doi:10.1109/18.904535

Cited by 7 publications

(18 citation statements)

References 14 publications

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“…For n ≥ 3, the similarity of C n (r, m) and B n (r, m) with RM codes runs deep. The likeness to RM codes was noted by Blackmore and Norton [6] for the case n being an odd prime. In the current work, we define the Berman and dual Berman codes using a recursive construction similar to the (u u u | u u u + v v v) Plotkin construction, and we show that their puncturing properties are similar to RM codes, they have a rich automorphism group (although they are not doubly transitive), they are generated by their minimum weight codewords, and that they can be decoded up to half the minimum distance efficiently.…”

Section: A Similarity With Reed-muller Codesmentioning

confidence: 57%

“…We begin this section with a review of abelian codes and DFT, and then introduce a convenient notation and present some results to work with the DFT for the group algebra F 2 [G m ], where G is an abelian group of order n. We then construct Berman codes and their duals as abelian codes and show that this construction is equivalent to our original recursive definition of these codes when n is odd. In Appendix II we show that the special case n being an odd prime coincides with the codes studied by Berman [5] and Blackmore and Norton [6]. For all odd n, we then identify a large family of abelian codes, that includes {B n (r, m) : n odd} and {C n (r, m) : n odd}, which achieves BEC capacity under bit-MAP decoding.…”

Section: B Overview Of the Main Results And Organizationmentioning

confidence: 81%

“…We study various basic properties of C n (r, m) and B n (r, m) in this work. A sub-class of these codes, corresponding to the case n = p with p being an odd prime, was studied by Berman [5], and Blackmore and Norton [6] using a group algebra framework. To the best of our knowledge, Berman [5] introduced and investigated the code B p (r, m) and showed that its minimum distance 2 r+1 is better than cyclic codes of the same length (for large values of m).…”

Section: Introductionmentioning

confidence: 99%

“…To the best of our knowledge, Berman [5] introduced and investigated the code B p (r, m) and showed that its minimum distance 2 r+1 is better than cyclic codes of the same length (for large values of m). Later, Blackmore and Norton [6] showed that the minimum distance of C p (r, m) is p m−r and analyzed the state complexity of this code. Blackmore and Norton refer to B p (r, m), the code originally designed by Berman in [5], as the Berman code.…”

Section: Introductionmentioning

confidence: 99%

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Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Natarajan¹,

Krishnan²

2022

Preprint

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We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as C n (r, m), and their duals are denoted as B n (r, m). The length of these codes is n m , where n ≥ 2, and r is their 'order'. When n = 2, C n (r, m) is the RM code of order r and length 2 m . The special case of these codes corresponding to n being an odd prime was studied by and . Following the terminology introduced by Blackmore and Norton, we refer to B n (r, m) as the Berman code and C n (r, m) as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group, they are generated by the minimum weight codewords, and that they can be decoded up to half the minimum distance efficiently. Using a result of Kumar et al. (2016), we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding. Furthermore, except double transitivity, they satisfy all the code properties used by Reeves and Pfister to show that RM codes achieve the capacity of binary-input memoryless symmetric channels. Finally, when n is odd, we identify a large class of abelian codes that includes B n (r, m) and C n (r, m) and which achieves BEC capacity.

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Section: A Similarity With Reed-muller Codesmentioning

confidence: 57%

Section: B Overview Of the Main Results And Organizationmentioning

confidence: 81%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Natarajan¹,

Krishnan²

2022

Preprint

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“…"Points of gain and fall" were introduced in [3,4,6,7] to help determine the state complexity of certain generalizations of Reed-Muller codes. For these codes, the points of gain and fall had particularly nice characterizations.…”

mentioning

confidence: 99%

Determining When the Absolute State Complexity of a Hermitian Code Achieves Its DLP Bound

Blackmore¹,

Norton²

2001

SIAM J. Discrete Math.

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Abstract. Let g be the genus of the Hermitian function field H/F q 2 and let C L (D, mQ∞) be a typical Hermitian code of length n. In [Des. Codes Cryptogr., to appear], we determined the dimension/length profile (DLP) lower bound on the state complexity of C L (D, mQ∞). Here we determine when this lower bound is tight and when it is not.For+ 2g, the DLP lower bounds reach Wolf's upper bound on state complexity and thus are trivially tight. We begin by showing that for about half of the remaining values of m the DLP bounds cannot be tight. In these cases, we give a lower bound on the absolute state complexity of C L (D, mQ∞), which improves the DLP lower bound.Next we give a "good" coordinate order for C L (D, mQ∞). With this good order, the state complexity of C L (D, mQ∞) achieves its DLP bound (whenever this is possible). This coordinate order also provides an upper bound on the absolute state complexity of C L (D, mQ∞) (for those values of m for which the DLP bounds cannot be tight). Our bounds on absolute state complexity do not meet for some of these values of m, and this leaves open the question whether our coordinate order is best possible in these cases.A straightforward application of these results is that if , and, in particular, so does its absolute state complexity.

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Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Natarajan

Krishnan

2022

2022 IEEE International Symposium on Information Theory (ISIT)

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We identify a family of binary codes whose structure is similar to Reed-Muller (RM) codes and which include RM codes as a strict subclass. The codes in this family are denoted as Cn(r, m), and their duals are denoted as Bn(r, m). The length of these codes is n m , where n ≥ 2, and r is their 'order'. When n = 2, Cn(r, m) is the RM code of order r and length 2 m . The special case of these codes corresponding to n being an odd prime was studied by Berman (1967) and Blackmore and Norton (2001). Following the terminology introduced by Blackmore and Norton, we refer to Bn(r, m) as the Berman code and Cn(r, m) as the dual Berman code. We identify these codes using a recursive Plotkin-like construction, and we show that these codes have a rich automorphism group. Applying a result of Kumar et al. (2016) to this set of automorphisms, we show that these codes achieve the capacity of the binary erasure channel (BEC) under bit-MAP decoding.

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On a family of Abelian codes and their state complexities

Cited by 7 publications

References 14 publications

Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

Determining When the Absolute State Complexity of a Hermitian Code Achieves Its DLP Bound

Berman Codes: A Generalization of Reed-Muller Codes that Achieve BEC Capacity

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