## Improved Upper Bounds on Stopping Redundancy (2007)

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Citations: | 23 - 3 self |

### BibTeX

@MISC{Han07improvedupper,

author = {Junsheng Han and Paul H. Siegel},

title = {Improved Upper Bounds on Stopping Redundancy},

year = {2007}

}

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### Abstract

For a linear block code with minimum distance d, its stopping redundancy is the minimum number of check nodes in a Tanner graph representation of the code, such that all nonempty stopping sets have size d or larger. We derive new upper bounds on stopping redundancy for all linear codes in general, and for maximum distance separable (MDS) codes specifically, and show how they improve upon previous results. For MDS codes, the new bounds are found by upper-bounding the stopping redundancy by a combinatorial quantity closely related to Turán numbers. (The Turán number, „

### Citations

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(Show Context)
Citation Context ...2. Also, note that for all MDS codes with minimum distance d, any set of d coordinates is the support of at least one codeword. These properties (and many more) can be found in MacWilliams and Sloane =-=[5]-=-. The authors of [1] noted the following. 1 Theorem 10 Let C be a MDS code with length n and minimum distance d. Then ρ(C) ≥ T(n, d − 1, d − 2). (23) Proof: Suppose H is a parity-check matrix for C an... |

1751 | The Probabilistic Method
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Citation Context ...Muller codes has been further studied by Etzion [3]. In this paper, we propose new upper bounds on ρ(C) and compare them to those in [1]. For the general upper bound, we take a probabilistic approach =-=[4]-=-. In the case of MDS codes, Schwartz and Vardy pointed out a a link between ρ(C) and covering numbers. This led to a number of lower bounds on ρ(C). We show that ρ(C) of MDS codes is upper bounded by ... |

1012 | Parity-Check Codes
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(Show Context)
Citation Context ... ≤ 1 − 2d−1 ) ρ d−1 ∑ ( ) n . (11) i Next, for 0 < δ = d(C) n d(C)−1 ∑ i=1 i=1 1 < 2 , it can be shown that ( n i ) < δ 1 − 2δ ✷ ( ) n . (12) δn Further, by Stirling’s approximation it is known that (=-=[6]-=-) ( ) n 1 ≤ √ 2 δn 2πnδ(1 − δ) nh(δ) . (13) Now, by putting together (11), (12), and (13), and referring to (4), we see that a positive solution to the equation 1 √ 2 2πnδ(1 − δ) nh(δ) ( ) ρ 1 − = 1. ... |

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Citation Context ...≥ 6 and minimum distance d = 4. Then ⌊ n ⌋ (⌈ n ⌉ ) ρ(C) = T(n, 3, 2) = − 1 . (31) 2 2 ✷ Proof: The formula for T(n, 3, 2) is a known result first discovered by Mantel [10] in 1907. Later, Turán [11] =-=[12]-=- solved the more general case of T(n, k, 2). It suffices to show that Γ(n, 2) ≤ T(n, 3, 2). Let T be a Turán (n, 3, 2)-system with smallest size. We show that T must also be a single-exclusion (n, 2)-... |

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Citation Context ...ell-studied. In fact, the determination of t(k, r) for k > r > 2 has been one of the most challenging open problems in combinatorial theory (for the solution of which Erdős offered a $1000 award; see =-=[15]-=-). Some of the known bounds on t(r+ 1, r) are summarized in Table I (cf. [11], [12], [8], [16], [17], [18], [19], [20]). 2Functions f(x) and g(x) are said to be asymptotic to each other as x → f(x) x0... |

44 | On the stopping distance and the stopping redundancy of codes
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(Show Context)
Citation Context ...ng redundancy of the code, denoted by ρ(C), represents the minimum number of rows in a paritycheck matrix such that s(H) = d(C). This quantity was defined and first investigated by Schwartz and Vardy =-=[1]-=-, [2]. They showed that ρ(C) is well-defined in that by proper choice of H, s(H) = d(C) can always be achieved. They then J. Han and P. H. Siegel are with the University of California, San Diego, La J... |

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Citation Context ..., t) covering design is a Turán (v, v −t, v −k)-system and vice versa. Hence, C(v, k, t) = T(v, v − t, v − k). For more information on covering designs and covering numbers, the reader is referred to =-=[9]-=-. ✷HAN AND SIEGEL: IMPROVED UPPER BOUNDS ON STOPPING REDUNDANCY 5 Let us denote this number by Γ ′ (n, d). Clearly, Γ ′ (n, d) is an upper bound of ρ(C). Note that Γ ′ (n, d) always exists since a ma... |

20 |
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Citation Context ... T(5, 3, 2) = 4. But it can be shown that ρ(C) = 5. ✷ For d = 5, we first note a couple of bounds on T(n, 4, 3). Lemma 16 Lemma 17 For n ≥ 13, T(n, 4, 3) ≥ 56 ( ) n . (33) 143 3 ✷ Proof: It is known (=-=[14]-=-) that T(n, k, r)/ ( ) n is nondecreasing in n, hence ( n0 r T(n, k, r) ≥ T(n0, k, r) ) ( ) n , for n ≥ n0. (34) r Since T(13, 4, 3) = 112 by Lemma 16, the result follows. Theorem 18 Let C be an MDS c... |

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17 |
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(Show Context)
Citation Context ...th n ≥ 6 and minimum distance d = 4. Then ⌊ n ⌋ (⌈ n ⌉ ) ρ(C) = T(n, 3, 2) = − 1 . (31) 2 2 ✷ Proof: The formula for T(n, 3, 2) is a known result first discovered by Mantel [10] in 1907. Later, Turán =-=[11]-=- [12] solved the more general case of T(n, k, 2). It suffices to show that Γ(n, 2) ≤ T(n, 3, 2). Let T be a Turán (n, 3, 2)-system with smallest size. We show that T must also be a single-exclusion (n... |

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Citation Context ...g open problems in combinatorial theory (for the solution of which Erdős offered a $1000 award; see [15]). Some of the known bounds on t(r+ 1, r) are summarized in Table I (cf. [11], [12], [8], [16], =-=[17]-=-, [18], [19], [20]). 2Functions f(x) and g(x) are said to be asymptotic to each other as x → f(x) x0 if limx→x0 = 1, and is denoted by f(x) ∼ g(x). In this paper g(x) we usually talk about integer fun... |

7 |
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(Show Context)
Citation Context ... The smallest number of blocks in a Turán (v, k, t)-system is known as the Turán number, and is correspondingly denoted by T(v, k, t). For more information on Turán numbers, the reader is referred to =-=[8]-=-, and references therein. Consider an MDS code C of length-n and minimum distance d. Then its dual code, C ⊥ , is an MDS code with minimum distance d ⊥ = n−d+2. Also, note that for all MDS codes with ... |

7 | Lower bounds for Turan's Problem - Frankl, Rödl - 1985 |

6 | An upper bound for the Turan number t3(n; 4
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(Show Context)
Citation Context ...lenging open problems in combinatorial theory (for the solution of which Erdős offered a $1000 award; see [15]). Some of the known bounds on t(r+ 1, r) are summarized in Table I (cf. [11], [12], [8], =-=[16]-=-, [17], [18], [19], [20]). 2Functions f(x) and g(x) are said to be asymptotic to each other as x → f(x) x0 if limx→x0 = 1, and is denoted by f(x) ∼ g(x). In this paper g(x) we usually talk about integ... |

5 |
Vraagstuk XXVIII,Wiskundige Opgaven met de Oplossingen 10
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- 1907
(Show Context)
Citation Context ...t C be a MDS code with length n ≥ 6 and minimum distance d = 4. Then ⌊ n ⌋ (⌈ n ⌉ ) ρ(C) = T(n, 3, 2) = − 1 . (31) 2 2 ✷ Proof: The formula for T(n, 3, 2) is a known result first discovered by Mantel =-=[10]-=- in 1907. Later, Turán [11] [12] solved the more general case of T(n, k, 2). It suffices to show that Γ(n, 2) ≤ T(n, 3, 2). Let T be a Turán (n, 3, 2)-system with smallest size. We show that T must al... |

5 | On asymmetric coverings and covering numbers - Applegate, Rains, et al. |

5 | On a problem of Turán - Kim, Roush - 1983 |

3 | Generic erasure correcting sets: Bounds and constructions - Hollmann, Tolhuizen - 2006 |

3 | On an extremal problem in graph theory - an, P - 1941 |

2 | Generating parity check equations for bounded-distance iterative erasure decoding - Hollmann, Tolhuizen - 2006 |

2 |
Extremal problems in the theory of graphs,” in Theory of Graphs and Their
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(Show Context)
Citation Context ... i, j} would not be covered. ✷ Fact 3: All blocks that are disjoint from ι form a Turán (n − Proof: The upper bound comes from a construction of 3, 4, 3)-system. Turán (n, 4, 3)-systems due to Ringel =-=[13]-=-, which has been Together, these imply that T(n, 4, 3) = |T | ≥ 1 + 2 verified to be optimal for n ≤ 13 ([9]). ( ) n−3 2 + T(n − 3, 4, 3), which contradicts Lemma 16 and Lemma 17 for n = 6, . . .,53. ... |

2 |
the stopping distance and stopping redundancy of codes
- “On
- 2005
(Show Context)
Citation Context ...near in n, and d(C) fixed). Example 1 Let G24 denote the extended binary Golay (24, 12, 8) code. In [1], it was shown by explicit construction that ρ(G24) ≤ 35. This was later improved to ρ(G24) ≤ 34 =-=[2]-=-. ✷ Applying the upper bounds obtained in this section to G24, we see that Theorem 1 gives ρ(G24) ≤ 2509, Theorem 2 gives ρ(G24) ≤ 1816, and Theorem 3 gives ρ(G24) ≤ 232. Also, the relaxed bounds in C... |

1 | Erdős on Graphs—His Legacy of Unsolved Problems - Chung, Graham - 1998 |

1 | An upper bound for the Turán number - Chung, Lu - 1999 |

1 |
Systems of sets that have the T-property
- Sidorenko
- 1981
(Show Context)
Citation Context ... problems in combinatorial theory (for the solution of which Erdős offered a $1000 award; see [15]). Some of the known bounds on t(r+ 1, r) are summarized in Table I (cf. [11], [12], [8], [16], [17], =-=[18]-=-, [19], [20]). 2Functions f(x) and g(x) are said to be asymptotic to each other as x → f(x) x0 if limx→x0 = 1, and is denoted by f(x) ∼ g(x). In this paper g(x) we usually talk about integer functions... |