Constant-Rank Codes and Their Connection to Constant-Dimension Codes

Gadouleau, Maximilien; Yan, Zhiyuan

doi:10.1109/tit.2010.2048447

Cited by 65 publications

(68 citation statements)

References 38 publications

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“…For these codes, hence, no polynomialtime list decoding can exist. The derivations apply connections between constant-rank codes and constant-dimension codes by Gadouleau and Yan [16]. Moreover, our results show that purely as a function of the length n and the minimum rank distance d, there cannot exist a polynomial upper bound similar to the Johnson bound in Hamming metric.…”

Section: Introductionmentioning

confidence: 78%

Bounds on polynomial-time list decoding of rank metric codes

Wachter-Zeh

2013

2013 IEEE International Symposium on Information Theory

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This contribution provides bounds on the list size of rank metric codes in order to understand whether polynomialtime list decoding is possible or not. First, an exponential upper bound is derived, which holds for any rank metric code of length n and minimum rank distance d. Second, a lower bound proves that there exists a rank metric code over Fqm of length n ≤ m such that the list size is exponential in the length of the code for any radius greater than half the minimum distance. This implies that in rank metric there cannot exist a polynomial upper bound depending only on n and d as the Johnson bound for Hamming metric. These bounds reveal significant differences between codes in Hamming and rank metric.

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Section: Introductionmentioning

confidence: 78%

Bounds on polynomial-time list decoding of rank metric codes

Wachter-Zeh

2013

2013 IEEE International Symposium on Information Theory

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“…A constant-rank code (CRC) is a rank metric code whose codewords have the same rank [9]. The maximum cardinality of a CRC in GF(q)mxn with minimum rank distance d and constant-rank r, denoted as AR(q, m, n, d, r), is studied in [9].…”

Section: A Rank Metric Codesmentioning

confidence: 99%

“…The maximum cardinality of a CRC in GF(q)mxn with minimum rank distance d and constant-rank r, denoted as AR(q, m, n, d, r), is studied in [9]. In particular, it is shown that A R (q, m, n, d, r) == ISIr 2009, Seoul, Korea, June 28 - July 3, 2009 AR(q, n, m, d, r) and, for n ::; m and d ::; r, AR(q,m,n,d,r) < [;]a:(m,r~d+l), (2) where a(m,u) == TI~==-ol(qm -qi).…”

Section: A Rank Metric Codesmentioning

confidence: 99%

“…First, CDCs can be constructed by lifting rank metric codes [8], which preserves the distance. Also, it was recently shown that CDCs are closely related to constant rank codes [9]. The maximum cardinality of a code with a given minimum rank distance was determined in [10]- [12].…”

Section: Introductionmentioning

confidence: 99%

“…Second, we show that DEP for KK codes is the highest up to a scalar among all CDCs obtained by lifting rank metric codes. Our study of the DEP of liftings of rank metric codes is based on constant-rank codes (CRCs) [9], a class of rank metric codes closely related to CDCs.…”

Section: Introductionmentioning

confidence: 99%

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Decoder error probability of bounded distance decoders for constant-dimension codes

Gadouleau

Yan

2009

2009 IEEE International Symposium on Information Theory

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Constant-dimension codes (CDCs) have been considered for error correction in random linear network coding, and low-complexity bounded distance decoders have been proposed. However, error performance, decoder error probability (DEP) in particular, of these bounded distance decoders has received little attention. In this paper, we first establish some fundamental geometric properties of the projective space. In particular, we show that the volume of the intersection of two spheres depends on only the two radii as well as the distance between and the dimensions of the two centers. Using these geometric properties, we then consider bounded distance decoders in both subspace and injection metrics and derive analytical expressions of their DEPs for CDCs over a symmetric operator channel, which ultimately depend on their distance distributions. Finally, we focus on CDCs obtained by lifting rank metric codes since their distance distributions are known, and obtain two important results. First, we obtain asymptotically tight upper bounds on the DEPs of bounded distance decoders in both metrics; the upper bounds decrease exponentially with the square of the minimum distance. Second, we show that the DEP for KK codes, obtained by lifting Gabidulin codes, is the highest up to a scalar among all CDCs obtained by lifting rank metric codes.

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