The Diametric Theorem in Hamming Spaces—Optimal Anticodes

Ahlswede, Rudolf; Khachatrian, Levon H.

doi:10.1006/aama.1998.0588

Cited by 89 publications

(47 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Notice that α (H q (n, d)), the independence number of the Hamming graph H q (n, d), is actually the maximum number of sequences such that the Hamming distance between any two of them is at most d − 1. Following [3], we define N q (n, s) to be the maximum number of q-ary sequences of length n that intersect pairwise (have the same entries) in at least s positions. It follows that…”

Section: Bounds On Codes Definition 4 (Hamming Graph [2])mentioning

confidence: 99%

See 1 more Smart Citation

Bounds on Codes Based on Graph Theory

Rouayheb

Georghiades

Soljanin³

et al. 2007

2007 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

Abstract-Let Aq(n, d) be the maximum order (maximum number of codewords) of a q-ary code of length n and Hamming distance at least d. And let A(n, d, w) that of a binary code of constant weight w. Building on results from algebraic graph theory and Erdős-ko-Rado like theorems in extremal combinatorics, we show how several known bounds on Aq(n, d) and A(n, d, w) can be easily obtained in a single framework. For instance, both the Hamming and Singleton bounds can derived as an application of a property relating the clique number and the independence number of vertex transitive graphs. Using the same techniques, we also derive some new bounds and present some additional applications.

show abstract

Section: Bounds On Codes Definition 4 (Hamming Graph [2])mentioning

confidence: 99%

“…The number N q (n, t) is well studied in extremal combinatorics [3] [5], and a closed form for it is known. Thus, exact expressions of N q (n, t) can be used to derive better upper bounds on A q (n, d).…”

Section: Proofmentioning

confidence: 99%

Bounds on Codes Based on Graph Theory

Rouayheb

Georghiades

Soljanin³

et al. 2007

2007 IEEE International Symposium on Information Theory

View full text Add to dashboard Cite

show abstract

“…Clearly we have w(n, p, 4, 36) ≤ w(n, p, 2, 36), and the Ahlswede-Khachatrian result [3] already shows (7) for p ≤ 1/(t + 1) = 1/37. We can easily improve this upper bound for p using (3).…”

Section: An Examplementioning

confidence: 72%

“…It might be natural to expect w(n, p, r,t) = max i w p (G i (n, r,t)). Ahlswede and Khachatrian proved that this is true for r = 2 in [3] (cf. [5,7,23]).…”

Section: N Rt) = {G ⊂ [N] : |G ∩ [T + Ri]| ≥ T + (R − 1)i} F I (Nmentioning

confidence: 88%