On the Feng-Rao Bound for Generalized Hamming Weights

Geil, Olav; Thommesen, Christian

doi:10.1007/11617983_29

Cited by 8 publications

(9 citation statements)

References 16 publications

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“…Even, it is seen to be of no significance that B should be a basis of F n q merely as just being some subset of F n q . For an outline of these observations we invite the reader to consult [12]. In the case of dual codes C ⊥ (B, G) the idea of using two bases B and B has proven fruitful in the study of cyclic codes.…”

Section: Definitionmentioning

confidence: 99%

“…The proof of the above version of the Feng-Rao bound can be found in [12]. The proof there uses many of the same ideas as does the proof of Theorem 10. In the examples at the end of the paper we will see that the two bounds sometimes give similar results and sometimes they do not.…”

Section: Definition 15mentioning

confidence: 99%

“…. , b n } is of the form (12). According to our agenda we should now be concerned with studying which ordered pairs (i, j ) ∈ I 2 that are well behaving and we should be concerned with deciding the valueρ(b i * b j ).…”

Section: Definition 24mentioning

confidence: 99%

“…The result concerning d(C(λ)) and d(C(δ)) is known as the order bound and comes from [15]. The result concerning d t (C(λ)) is from [13] and the result concerning d t (C(δ)) is from [12]. The results concerning E(λ) andẼ(λ) are new.…”

Section: Proposition 27 Letmentioning

confidence: 99%

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Evaluation codes from order domain theory

Andersen

Geil

2008

Finite Fields and Their Applications

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The celebrated Feng-Rao bound estimates the minimum distance of codes defined by means of their parity check matrices. From the Feng-Rao bound it is clear how to improve a large family of codes by leaving out certain rows in their parity check matrices. In this paper we derive a simple lower bound on the minimum distance of codes defined by means of their generator matrices. From our bound it is clear how to improve a large family of codes by adding certain rows to their generator matrices. The new bound is very much related to the Feng-Rao bound as well as to Shibuya and Sakaniwa's bound in [T. Shibuya, K. Sakaniwa, A dual of well-behaving type designed minimum distance, IEICE Trans. Fund. E84-A (2001) 647-652]. Our bound is easily extended to deal with any generalized Hamming weights. We interpret our methods into the setting of order domain theory. In this way we fill in an obvious gap in the theory of order domains.

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Section: Definitionmentioning

confidence: 99%

Section: Definition 15mentioning

confidence: 99%

Section: Definition 24mentioning

confidence: 99%

Section: Proposition 27 Letmentioning

confidence: 99%

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Evaluation codes from order domain theory

Andersen

Geil

2008

Finite Fields and Their Applications

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“…Building on [16,18,35] Andersen and Geil in [1] introduced a bound on the minimum distance of primary codes. This bound was later slightly generalized and enhanced in [23], but we shall refer also this version as Andersen and Geil's bound. The bound has the same flavor as the Feng-Rao bound.…”

Section: Introductionmentioning

confidence: 99%

Feng–Rao decoding of primary codes

Geil

Matsumoto

Ruano

2013

Finite Fields and Their Applications

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We show that the Feng-Rao bound for dual codes and a similar bound by Andersen and Geil [1] for primary codes are consequences of each other. This implies that the Feng-Rao decoding algorithm can be applied to decode primary codes up to half their designed minimum distance. The technique applies to any linear code for which information on well-behaving pairs is available. Consequently we are able to decode efficiently a large class of codes for which no non-trivial decoding algorithm was previously known. Among those are important families of multivariate polynomial codes. Matsumoto and Miura in [30] (See also [3]) derived from the Feng-Rao bound a bound for primary one-point algebraic geometric codes and showed how to decode up to what is guaranteed by their bound. The exposition in [30] requires the use of differentials which was not needed in [1]. Nevertheless we demonstrate a very strong connection between Matsumoto and Miura's bound and Andersen and Geil's bound when applied to primary one-point algebraic geometric codes.

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