## Upper Bounds for Constant-Weight Codes (2000)

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Venue: | IEEE TRANS. INFORM. THEORY |

Citations: | 31 - 1 self |

### BibTeX

@ARTICLE{Agrell00upperbounds,

author = {Erik Agrell and Alexander Vardy and Kenneth Zeger},

title = {Upper Bounds for Constant-Weight Codes},

journal = {IEEE TRANS. INFORM. THEORY},

year = {2000},

volume = {46},

pages = {2373--2395}

}

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### Abstract

Let A(n; d; w) denote the maximum possible number of codewords in an (n; d; w) constant-weight binary code. We improve upon the best known upper bounds on A(n; d; w) in numerous instances for n 6 24 and d 6 12, which is the parameter range of existing tables. Most improvements occur for d = 8; 10, where we reduce the upper bounds in more than half of the unresolved cases. We also extend the existing tables up to n 6 28 and d 6 14. To obtain these results, we develop new techniques and introduce new classes of codes. We derive a number of general bounds on A(n; d; w) by means of mapping constantweight codes into Euclidean space. This approach produces, among other results, a bound on A(n; d; w) that is tighter than the Johnson bound. A similar improvement over the best known bounds for doubly-constant-weight codes, studied by Johnson and Levenshtein, is obtained in the same way. Furthermore, we introduce the concept of doubly-boundedweight codes, which may be thought of as a generaliz...

### Citations

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Citation Context ...ystems, radar and sonar signal design, mobile radio, and synchronization [9, 11, 19]. For general background on constant-weight codes, and the related class of spherical codes, we refer the reader to =-=[22, 28, 42]-=-. The rest of this paper is organized as follows. In the next section, we define concepts and terminology that will be used throughout this work. A simple mapping from binary codes to spherical codes ... |

421 | The Theory and Practice of Error Control Codes - Blahut - 1984 |

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Citation Context ...cannot equal 272 either. Hence A(15; 4; 5) 6 271, which was stated without proof in [8] (though A(15; 4; 5) 6 272 was proved there). 2 B. The Freiman-Berger-Johnson Bound The well-known Hamming bound =-=[33]-=- for unrestricted codes is obtained by centering a sphere around each codeword. Johnson [37] developed a family of bounds for constant-weight codes using a similar technique, and thereby generalized a... |

230 |
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Citation Context ...s simple idea often leads to powerful upper bounds on A(n; d; w) (cf. Examples 2, 3, and 4). Finally, as in most previous work on the subject, we make use of linear programming, based on the Delsarte =-=[24]-=- inequalities for constant-weight codes. It is known that the distance distribution of constant-weight codes is subject to more constraints than can be obtained from Delsarte inequalities, but determi... |

163 | World-System Theory - Chirot, Hall - 1982 |

100 | New upper bounds on the rate of a code via the Delsarte-MacWilliams inequalities
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Citation Context ...t codes and bounds on doubly-bounded-weight codes. hand side of (1) can be reduced by a factor of two. The best known asymptotic upper bound on A(n; d), given by McEliece, Rodemich, Rumsey, and Welch =-=[43]-=- in 1977, consists of this inequality in conjunction with a linear programming bound on the size of constant-weight codes. Thus it should not be surprising that better bounds on A(n; d; w) lead to new... |

86 | W.D.: A new table of constant weight codes
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Citation Context ... in 1980 by Graham and Sloane [31]. Then in 1990, following a large number of new explicit code constructions for certain parameters, came the encyclopedic work of Brouwer, Shearer, Sloane, and Smith =-=[17]-=-, where the best known lower bounds on A(n; d; w) for n 6 28 and d 6 18 are collected. Upper bounds are given in [17] only for those parameters where these bounds are known to coincide with the lower ... |

60 | The search for a finite projective plane of order 10
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Citation Context ...es in Theorem 21. In general, it is very difficult to check specific upper bounds found by others. (As pointed out in [17], an extreme case of this is the celebrated result of Lam, Thiel, and Swiercz =-=[38]-=- that there is no projective plane of order 10, which is equivalent to A(111; 20; 11) 6 110. The proof of [38] is based on years of research and thousands of hours of computer time.) Thus Theorem 21 r... |

57 |
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Citation Context ... they are generally recognized as an important class of codes in their own right. They have been recently introduced in a number of engineering applications, including CDMA systems for optical fibers =-=[19]-=-, protocol design for the collision channel without feedback [1], automatic-repeat-request error control systems [54], and parallel asynchronous communication [12]. In addition, they often serve as bu... |

54 |
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Citation Context ...on A(n; d; w) mentioned in the foregoing paragraph, we develop a number of new general approaches to the problem. Some of these are briefly described below. It is well known since the work of Johnson =-=[35]-=- and Levenshtein [39] that certain bounds on A(n; d; w) can be derived using doubly-constantweight codes, which constitute a special restricted sub-class of constant-weight codes. In this work, we int... |

48 |
Lower bounds for constant weight codes
- Graham, Sloane
- 1980
(Show Context)
Citation Context ...then, there has been very little progress on the upper bounds. In contrast, lower bounds on A(n; d; w) were improved upon many times. The lower bounds of [8] were revised in 1980 by Graham and Sloane =-=[31]-=-. Then in 1990, following a large number of new explicit code constructions for certain parameters, came the encyclopedic work of Brouwer, Shearer, Sloane, and Smith [17], where the best known lower b... |

48 |
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- 1950
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Citation Context ...em 13 holds, and the theorem proves that cannot equal either. Hence , which was stated without proof in [8] (though was proved there). B. The Freiman–Berger–Johnson Bound The well-known Hamming bo=-=und [33]-=- for unrestricted codes is obtained by centering a sphere around each codeword. Johnson [37] developed a family of bounds for constant-weight codes using a similar technique, and thereby generalized a... |

36 |
Binary codes with specified minimum distance
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Citation Context ...(29) T (w 1 ; n 1 ; w 2 ; n 2 ; 2ffi) 6 2n 1 + 2n 2 \Gamma 4; if b = 0 (30) where b = ffi \Gamma w 1 (n 1 \Gamma w 1 ) n 1 \Gamma w 2 (n 2 \Gamma w 2 ) n 2 Corollary 4 is similar to the Plotkin bound =-=[44]-=-. The only difference is that in the latter, the right-hand side of (23) is truncated to an even value, instead of just an integer as in Corollary 4. Hence the Plotkin bound is stronger. It was derive... |

35 | Handbook of Incidence Geometry - Buekenhout - 1995 |

33 |
On bounds for packings in n-dimensional Euclidean space, Doklady Akad
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- 1979
(Show Context)
Citation Context ...e we have found Theorem 31 to be most useful; through Theorem 30 and one of the paths in Figure 1, numerous upper bounds on A(n; d; w) were improved. For s ? 0, we have used Levenshtein's upper bound =-=[40]-=-, which resulted in some additional improvements for T 0 (w 1 ; n 1 ; w 2 ; n 2 ; d) at the expense of higher complexity. However, these improvements did not propagate to A(n; d; w) or T (w 1 ; n 1 ; ... |

33 |
The closest packing of spherical caps in n dimensions
- Rankin
- 1955
(Show Context)
Citation Context ...]. For s 6 0, this function is known exactly. Specifically, it is known that AS (n; s) = 1 \Gamma 1 s ; if s 6 \Gamma 1 n (11) AS (n; s) = n + 1; if \Gamma 1 n 6 s ! 0 (12) AS (n; 0) = 2n (13) Rankin =-=[47]-=- was the first to establish (11), while (12) was originally stated by Davenport and Haj'os [23], and proved by Acz'el and Szele [2]. Equation (13) was first stated by Erdos [26], and proved by Sarkadi... |

22 | Bounds for binary codes of length less than 25
- Best, Brouwer, et al.
- 1978
(Show Context)
Citation Context ... 1977 in the book of MacWilliams and Sloane [42, pp. 684--691], for n 6 24 and d 6 10. An updated version of these tables, along with a more complete treatment of the underlying theory, was published =-=[8]-=- in 1978. Another update appeared in Honkala's Licentiate thesis [34, Section 6], together with a new table of upper bounds for d = 12 and n 6 27. Since then, there has been very little progress on th... |

14 |
The CRC Handbook of Combinatorial Designs. Boca
- Colbourn, Dinitz
- 1996
(Show Context)
Citation Context ...hat it can be proved using a similar technique. The bounds (74) and (78) follow from the nonexistence of certain Steiner systems, while (86) and (96) follow from the nonexistence of certain 2-designs =-=[21, 25, 32]-=- (see [17] and the discussion following Theorem 12). These four bounds can each be decreased by one using Theorems 11 or 13. The value in (84) was derived in [13] from the nonexistence of a certain in... |

13 |
Combinatorial properties of group divisible incomplete block designs
- Bose, Connor
- 1952
(Show Context)
Citation Context ...ructure is known, in the terminology of design theory, as a group divisible incomplete block design with parameterss(v; r; b; k; m; n;s1 ;s2 ) = (22; 5; 22; 5; 11; 2; 0; 1), but no such design exists =-=[14]-=-. E. Redundant Bounds Many bounds for constant-weight codes have been proposed, but not all of them remain competitive today. Our intent in this work is to list all the upper bounds for constant-weigh... |

11 | On the optimum of Delsarte’s linear program
- Samorodnitsky
- 2001
(Show Context)
Citation Context ...estricted codes to constant-weight codes. It yields, in conjunction with the linear programming bound of [43], the best known upper bound on A(n; d; w) asymptotically, as n, d, and w tend to infinity =-=[4, 5, 48]-=-. Nevertheless, neither [39, eq.(4)] nor its strengthened version [39, eq.(5)] improve on any of the values in our tables. Neither does [41, Theorem 6.25], which has the same asymptotical performance.... |

10 | Constructions of binary constant-weight cyclic codes and cyclically permutable codes
- A, GyM, et al.
- 1992
(Show Context)
Citation Context ... [10, p. 451, 456]) and Bassalygo [6]. This elegant Bassalygo-Elias inequality A(n; d) 6 2 n \Gamma n w \Delta A(n; d; w) (1) was improved upon by Levenshtein [39, eq.(32)], and later by van Pul (see =-=[1]-=-), who pointed out that the rightPSfrag replacements A T T 0 9,10,13 11 14 20 24,28,29 22,27 25 26 32,35 33,34 36 26 Figure 1. The interdependence between bounds on the three types of binary constant-... |

10 |
Upper Bound Estimates For Fixed-Weight Codes,” Probl
- Levenshtein
- 1974
(Show Context)
Citation Context ...ed in the foregoing paragraph, we develop a number of new general approaches to the problem. Some of these are briefly described below. It is well known since the work of Johnson [35] and Levenshtein =-=[39]-=- that certain bounds on A(n; d; w) can be derived using doubly-constantweight codes, which constitute a special restricted sub-class of constant-weight codes. In this work, we introduce the concept of... |

10 |
On error-correcting balanced codes
- Tilborg, Blaum
- 1989
(Show Context)
Citation Context ...-request error control systems [54], and parallel asynchronous communication [12]. In addition, they often serve as building blocks in the design of spherical codes [28] and DC-free constrained codes =-=[29, 52]-=-. Further applications have been reported in frequency-hopping spread-spectrum systems, radar and sonar signal design, mobile radio, and synchronization [9, 11, 19]. For general background on constant... |

10 |
New upper bounds for error correcting codes, Problemy Peredachi Informatsii 1
- Bassalygo
- 1965
(Show Context)
Citation Context ...two of them differ in at least positions; the maximum number of codewords in any such code is usually denoted .An important relation between and is due to Elias (see [10, pp. 451, 456]) and Bassalygo =-=[6]. Th-=-is elegant Fig. 1. The interdependence between bounds on the three types of binary constant-weight codes: e stands for general constant-weight codes, „ for doubly-bounded-weight codes, and „ for d... |

10 | Mathematics Handbook for Science and Engineering - R˚ade, Westergren - 2004 |

9 |
Codes on Euclidean Spheres
- Ericson, Zinoviev
- 2001
(Show Context)
Citation Context ...hout feedback [1], automatic-repeat-request error control systems [54], and parallel asynchronous communication [12]. In addition, they often serve as building blocks in the design of spherical codes =-=[28]-=- and DC-free constrained codes [29, 52]. Further applications have been reported in frequency-hopping spread-spectrum systems, radar and sonar signal design, mobile radio, and synchronization [9, 11, ... |

7 | On the upper bounds for unrestricted binary errorcorrecting codes - Johnson - 1971 |

6 |
Constant weight codes and group divisible design
- Blake-Wilson, Phelps
- 1999
(Show Context)
Citation Context ...des [28] and DC-free constrained codes [29, 52]. Further applications have been reported in frequency-hopping spread-spectrum systems, radar and sonar signal design, mobile radio, and synchronization =-=[9, 11, 19]-=-. For general background on constant-weight codes, and the related class of spherical codes, we refer the reader to [22, 28, 42]. The rest of this paper is organized as follows. In the next section, w... |

6 |
Combinatorial bounds for binary constant weight and covering codes, Ph
- Honkala
- 1989
(Show Context)
Citation Context ...m are three-fold, as described in the next three paragraphs. First, we improve upon the existing upper bounds on A(n; d; w) in many instances. For example, out of the 23 unresolved cases for d = 8 in =-=[17, 34]-=-, 14 upper bounds are improved upon in this paper. For d = 10, we update 10 out of the 18 unresolved cases. As a result, we establish seven new exact values of A(n; d; w), and re-derive by analytical ... |

6 | bounds for codes and designs, Handbook of Coding Theory - Universal - 1998 |

5 |
On relations between covering radius and dual distance
- Ashkihmin, Honkala, et al.
- 1999
(Show Context)
Citation Context ...estricted codes to constant-weight codes. It yields, in conjunction with the linear programming bound of [43], the best known upper bound on A(n; d; w) asymptotically, as n, d, and w tend to infinity =-=[4, 5, 48]-=-. Nevertheless, neither [39, eq.(4)] nor its strengthened version [39, eq.(5)] improve on any of the values in our tables. Neither does [41, Theorem 6.25], which has the same asymptotical performance.... |

5 |
Upper bounds on the minimum distance of spherical codes
- Boyvalenkov, Danev, et al.
- 1996
(Show Context)
Citation Context ..., the best known general upper bound on AS (n; s) was given by Levenshtein in [40]. This bound can be improved upon for certain specific parameters using the methods of Boyvalenkov, Danev, and Bumova =-=[15]-=-. For s 6 0, this function is known exactly. Specifically, it is known that AS (n; s) = 1 \Gamma 1 s ; if s 6 \Gamma 1 n (11) AS (n; s) = n + 1; if \Gamma 1 n 6 s ! 0 (12) AS (n; 0) = 2n (13) Rankin [... |

5 |
Spherical codes generated by binary partitions of symmetric pointsets
- Ericson, Zinoviev
- 1995
(Show Context)
Citation Context ...erical codes. Conversely, an upper bound on the cardinality of spherical codes serves as an upper bound for binary codes. The former relation has been successfully exploited --- see [22, pp. 26--27], =-=[27]-=-, [28] and references therein. One contribution of the present paper is to investigate the latter relation, from which we obtain improved bounds in some cases. This approach, which has been less highl... |

5 |
Upper bounds for constant weight error-correcting codes
- Johnson
- 1972
(Show Context)
Citation Context ..., which, in turn, are a sub-class of unrestricted codes. Unrestricted codes and constant-weight codes have been studied extensively in the past. Doubly-constant-weight codes were proposed in [39] and =-=[37]-=-. The class of doublybounded -weight codes is introduced in this paper; it turns out to be very useful in deriving bounds for the other classes. In the following, d(C ) denotes the minimum Hamming dis... |

5 |
personal communication
- Shearer
- 1997
(Show Context)
Citation Context ...es later should be "[3, Table IIIA]". In [17, Table III], "A(23; 10; 7) = 21" should be "A(23; 10; 7) = 20" and the corresponding entries in [17, Tables I--D and XVI] sho=-=uld give 20 as an exact value [49]-=-. The value A(21; 10; 8) = 21 in [17, Table I--D] is not explained in [17, Table III]. It appears possible that [17, Table I--D] was wrong in stating that the value for A(21; 10; 8) was exact rather t... |

4 | New upper bounds on generalized weights
- Ashikhmin, Barg, et al.
- 1999
(Show Context)
Citation Context ...estricted codes to constant-weight codes. It yields, in conjunction with the linear programming bound of [43], the best known upper bound on A(n; d; w) asymptotically, as n, d, and w tend to infinity =-=[4, 5, 48]-=-. Nevertheless, neither [39, eq.(4)] nor its strengthened version [39, eq.(5)] improve on any of the values in our tables. Neither does [41, Theorem 6.25], which has the same asymptotical performance.... |

4 | Constructions for optimal constant weight cyclically permutable codes and difference families - Bitan, Etzion - 1995 |

4 |
On the structure of balanced incomplete block designs
- Connor
- 1952
(Show Context)
Citation Context ...hat it can be proved using a similar technique. The bounds (74) and (78) follow from the nonexistence of certain Steiner systems, while (86) and (96) follow from the nonexistence of certain 2-designs =-=[21, 25, 32]-=- (see [17] and the discussion following Theorem 12). These four bounds can each be decreased by one using Theorems 11 or 13. The value in (84) was derived in [13] from the nonexistence of a certain in... |

4 |
On upper bounds for error detecting and error-correcting codes of t-mite length,“IEEE
- Wax
(Show Context)
Citation Context ...me cases. This approach, which has been less highlighted than its converse, was used in [27] to prove two well-known bounds; see below in Section III-B. A somewhat related method was suggested by Wax =-=[55]-=-, who derived upper bounds 1 on binary codes from some sphere packings (not spherical codes) in Euclidean space. A. Binary Codes as Spherical Codes We first map three of the classes of binary codes in... |

3 |
Constructions of error-correcting DC-free block codes
- Etzion
- 1990
(Show Context)
Citation Context ...-request error control systems [54], and parallel asynchronous communication [12]. In addition, they often serve as building blocks in the design of spherical codes [28] and DC-free constrained codes =-=[29, 52]-=-. Further applications have been reported in frequency-hopping spread-spectrum systems, radar and sonar signal design, mobile radio, and synchronization [9, 11, 19]. For general background on constant... |

3 | Combinatorial Configurations. Designs, Codes, Graphs - Tonchev - 1988 |

3 | Pul, “On bounds on codes - van - 1982 |

2 |
Some additional upper bounds for fixed-weight codes of specified minimum distance
- Berger
- 1967
(Show Context)
Citation Context ... codes is obtained by centering a sphere around each codeword. Johnson [37] developed a family of bounds for constant-weight codes using a similar technique, and thereby generalized a bound by Berger =-=[7]-=-, who in turn generalized a bound by Freiman [30]. Johnson [37] gives a range of versions of the same general bound, which leaves the user of these bounds some freedom to choose a suitable level of co... |

2 |
Determination of a packing number
- Stinson
- 1977
(Show Context)
Citation Context ...f computer time.) Thus Theorem 21 relies on the published literature. We now provide references for each bound listed in Theorem 21. The bounds (77) and (79) were obtained by Brouwer [16] and Stinson =-=[51]-=-, respectively. The method used was assuming the existence of a code with a higher value of A(n; d; w), identifying properties of this hypothetical code, 12 June 24, 2000 and arriving at a contradicti... |

2 |
On the undetected error probability of nonlinear binary constant weight codes
- Wang, Yang
- 1994
(Show Context)
Citation Context ...in a number of engineering applications, including CDMA systems for optical fibers [19], protocol design for the collision channel without feedback [1], automatic-repeat-request error control systems =-=[54]-=-, and parallel asynchronous communication [12]. In addition, they often serve as building blocks in the design of spherical codes [28] and DC-free constrained codes [29, 52]. Further applications have... |

2 |
bounds for constant weight error correcting codes
- “Upper
- 1972
(Show Context)
Citation Context ...s, which, in turn, are a subclass of unrestricted codes. Unrestricted codes and constant-weight codes have been studied extensively in the past. Doubly-constant-weight codes were proposed in [39] and =-=[37]-=-. The class of doubly-bounded-weight codes is introduced in this paper; it turns out to be very useful in deriving bounds for the other classes. (3) (4) In the following, denotes the minimum Hamming d... |

1 |
Solution 35
- Szele
- 1952
(Show Context)
Citation Context ...; s) = n + 1; if \Gamma 1 n 6 s ! 0 (12) AS (n; 0) = 2n (13) Rankin [47] was the first to establish (11), while (12) was originally stated by Davenport and Haj'os [23], and proved by Acz'el and Szele =-=[2]-=-. Equation (13) was first stated by Erdos [26], and proved by Sarkadi and Szele [50]. Example 1. We have AS (25; \Gamma3=41) = b44=3c = 14 (to be continued in Example 15). 2 We now introduce the class... |

1 |
Tables of binary block codes," available online at www.chl.chalmers.se/~agrell
- Agrell, Vardy, et al.
(Show Context)
Citation Context ...eciate hearing of any improvements to the tables. To conserve space, our tables of upper bounds for T 0 (w 1 ; n 1 ; w 2 ; n 2 ; d) and T (w 1 ; n 1 ; w 2 ; n 2 ; d) are published electronically only =-=[3]-=-. On the same web site [3], we will also keep record of any updates or corrections that are brought to our attention. Most of the theorems in this paper yield upper bounds that depend on A(n; d; w), T... |

1 |
New upper bounds for error correcting codes," Probl
- Bassalygo
- 1968
(Show Context)
Citation Context ...st d positions; the maximum number of codewords in any such code is usually denoted A(n; d). An important relation between A(n; d) and A(n; d; w) is due to Elias (see [10, p. 451, 456]) and Bassalygo =-=[6]-=-. This elegant Bassalygo-Elias inequality A(n; d) 6 2 n \Gamma n w \Delta A(n; d; w) (1) was improved upon by Levenshtein [39, eq.(32)], and later by van Pul (see [1]), who pointed out that the rightP... |

1 |
Coding for tolerance and detection of skew in parallel asynchronous communications
- Blaum, Bruck
- 2000
(Show Context)
Citation Context ...ding CDMA systems for optical fibers [19], protocol design for the collision channel without feedback [1], automatic-repeat-request error control systems [54], and parallel asynchronous communication =-=[12]-=-. In addition, they often serve as building blocks in the design of spherical codes [28] and DC-free constrained codes [29, 52]. Further applications have been reported in frequency-hopping spread-spe... |