Perfect codes in the discrete simplex
We study the problem of existence of (nontrivial) perfect codes in the discrete n -simplex Δ ℓ n : = x 0 , … , x n : x i ∈ Z + , ∑ i x i = ℓ under ℓ 1 metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for er...
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| Vydané v: | Designs, codes, and cryptography Ročník 75; číslo 1; s. 81 - 95 |
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| Hlavní autori: | , |
| Médium: | Journal Article |
| Jazyk: | English |
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01.04.2015
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| ISSN: | 0925-1022, 1573-7586 |
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| Abstract | We study the problem of existence of (nontrivial) perfect codes in the discrete
n
-simplex
Δ
ℓ
n
:
=
x
0
,
…
,
x
n
:
x
i
∈
Z
+
,
∑
i
x
i
=
ℓ
under
ℓ
1
metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for error correction in the permutation channels. It is shown that
e
-perfect codes in the 1-simplex
Δ
ℓ
1
exist for any
ℓ
≥
2
e
+
1
, the 2-simplex
Δ
ℓ
2
admits an
e
-perfect code if and only if
ℓ
=
3
e
+
1
, while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets. |
|---|---|
| AbstractList | We study the problem of existence of (nontrivial) perfect codes in the discrete
n
-simplex
Δ
ℓ
n
:
=
x
0
,
…
,
x
n
:
x
i
∈
Z
+
,
∑
i
x
i
=
ℓ
under
ℓ
1
metric. The problem is motivated by the so-called multiset codes, which have recently been introduced by the authors as appropriate constructs for error correction in the permutation channels. It is shown that
e
-perfect codes in the 1-simplex
Δ
ℓ
1
exist for any
ℓ
≥
2
e
+
1
, the 2-simplex
Δ
ℓ
2
admits an
e
-perfect code if and only if
ℓ
=
3
e
+
1
, while there are no perfect codes in higher-dimensional simplices. In other words, perfect multiset codes exist only over binary and ternary alphabets. |
| Author | Vukobratović, Dejan Kovačević, Mladen |
| Author_xml | – sequence: 1 givenname: Mladen surname: Kovačević fullname: Kovačević, Mladen email: kmladen@uns.ac.rs organization: Department of Electrical Engineering, University of Novi Sad – sequence: 2 givenname: Dejan surname: Vukobratović fullname: Vukobratović, Dejan organization: Department of Electrical Engineering, University of Novi Sad |
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| Cites_doi | 10.1007/BF01390767 10.1007/s10623-009-9273-3 10.1007/978-1-4615-6666-3 10.1016/S0019-9958(75)80005-2 10.1007/BF01390772 |
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| Keywords | Integer codes 94B25 05C12 68R99 Discrete simplex 05B40 Sphere packing Multiset codes Manhattan metric 52C17 Permutation channel Perfect codes |
| Language | English |
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| References | Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. Elsevier, Amsterdam (1997). Gadouleau M., Goupil A.: Binary codes for packet error and packet loss correction in store and forward. In: Proceedings of the International ITG Conference on Source and Channel Coding, Siegen, Germany (2010) KötterR.KschischangF.R.Coding for errors and erasures in random network codingIEEE Trans. Inf. Theory200854835793591 GordonD.M.Perfect single error-correcting codes in the Johnson schemeIEEE Trans. Inf. Theory2006521046704672 HorakP.Tilings in Lee metricEur. J. Comb.2009302480489 ChiharaL.On the zeros of the Askey–Wilson polynomials, with applications to coding theorySIAM J. Math. Anal.1987181191207 ZinovievV.A.LeontievV.K.The nonexistence of perfect codes over Galois fieldsProbl. Control Inf. Theory19732123132 EtzionT.VardyA.Perfect binary codes: constructions, properties, and enumerationIEEE Trans. Inf. Theory1994403754763 Kovačević M., Vukobratović D.: Multiset codes for permutation channels. Available online at: arXiv:1301.7564. Bange D.W., Barkauskas A.E., Slater P.J.: Efficient dominating sets in graphs. In: Ringeisen R.D., Roberts F.S. (eds.) Applications of Discrete Mathematics, pp. 189–199. SIAM, Philadelphia (1988). EtzionT.Configuration distribution and designs of codes in the Johnson schemeJ. Comb. Des.20071511534 RoosC.A note on the existence of perfect constant weight codesDiscret. Math.198347121123 AlBdaiwiB.HorakP.MilazzoL.Enumerating and decoding perfect linear Lee codesDes. Codes Cryptogr.2009522155162 MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977). PostK.A.Nonexistence theorem on perfect Lee codes over large alphabetsInf. Control1975294369380 EtzionT.Product constructions for perfect Lee codesIEEE Trans. Inf. Theory2011571174737481 ŠpacapanS.Non-existence of face-to-face four dimensional tiling in the Lee metricEur. J. Comb.2007281127133 Aigner M.: Combinatorial Theory. Springer, New York (1979). MartinW.J.ZhuX.J.Anticodes for the Grassmann and bilinear forms graphsDes. Codes Cryptogr.1995617379 LevenshteinV.I.On perfect codes in deletion and insertion metricDiscret. Math. Appl.199223241258 Bertsekas D.P., Gallager R.: Data Networks, 2nd edn. Prentice Hall, Englewood Cliffs (1992). GolombS.W.WelchL.R.Perfect codes in the Lee metric and the packing of polyominoesSIAM J. Appl. Math.1970182302317 van Lint J. H.: Nonexistence theorems for perfect error-correcting codes. In: Computers in Algebra and Number Theory, vol. IV, SIAM-AMS Proceedings (1971). EtzionT.On the nonexistence of perfect codes in the Johnson schemeSIAM J. Discret. Math.199692201209 KovačevićM.VukobratovićD.Subset codes for packet networksIEEE Commun. Lett.2013174729732 BoursP.A.H.On the construction of perfect deletion-correcting codes using design theoryDes. Codes Cryptogr.199561520 EtzionT.VardyA.Error-correcting codes in projective spaceIEEE Trans. Inf. Theory201157211651173 BestM.R.Perfect codes hardly existIEEE Trans. Inf. Theory1983293349351 BiggsN.Perfect codes in graphsJ. Comb. Theory B1973153289296 TietäväinenA.On the nonexistence of perfect codes over finite fieldsSIAM J. Appl. Math.19732418896 HorakP.On perfect Lee codesDiscret. Math.20093091855515561 DelsarteP.An algebraic approach to association schemes and coding theoryPhilips J. Res.197310197 GadouleauM.GoupilA.A matroid framework for noncoherent random network communicationsIEEE Trans. Inf. Theory201157210311045 AstolaJ.On perfect Lee codes over small alphabets of odd cardinalityDiscret. Appl. Math.19824227228 ShimabukuroO.On the nonexistence of perfect codes in $$ J(2w + p2, w)$$ J ( 2 w + p 2 , w )Ars Comb.200575129134 EtzionT.SchwartzM.Perfect constant-weight codesIEEE Trans. Inf. Theory200450921562165 9893_CR20 9893_CR24 9893_CR23 9893_CR22 9893_CR21 9893_CR28 9893_CR27 9893_CR26 9893_CR9 9893_CR25 9893_CR8 9893_CR7 9893_CR6 9893_CR5 9893_CR29 9893_CR4 9893_CR3 9893_CR2 9893_CR1 9893_CR31 9893_CR30 9893_CR13 9893_CR35 9893_CR12 9893_CR34 9893_CR11 9893_CR33 9893_CR10 9893_CR32 9893_CR17 9893_CR16 9893_CR15 9893_CR14 9893_CR36 9893_CR19 9893_CR18 |
| References_xml | – reference: AlBdaiwiB.HorakP.MilazzoL.Enumerating and decoding perfect linear Lee codesDes. Codes Cryptogr.2009522155162 – reference: EtzionT.VardyA.Perfect binary codes: constructions, properties, and enumerationIEEE Trans. Inf. Theory1994403754763 – reference: EtzionT.Product constructions for perfect Lee codesIEEE Trans. Inf. Theory2011571174737481 – reference: EtzionT.SchwartzM.Perfect constant-weight codesIEEE Trans. Inf. Theory200450921562165 – reference: ChiharaL.On the zeros of the Askey–Wilson polynomials, with applications to coding theorySIAM J. Math. Anal.1987181191207 – reference: KötterR.KschischangF.R.Coding for errors and erasures in random network codingIEEE Trans. Inf. Theory200854835793591 – reference: GordonD.M.Perfect single error-correcting codes in the Johnson schemeIEEE Trans. Inf. Theory2006521046704672 – reference: RoosC.A note on the existence of perfect constant weight codesDiscret. Math.198347121123 – reference: Gadouleau M., Goupil A.: Binary codes for packet error and packet loss correction in store and forward. In: Proceedings of the International ITG Conference on Source and Channel Coding, Siegen, Germany (2010) – reference: GadouleauM.GoupilA.A matroid framework for noncoherent random network communicationsIEEE Trans. Inf. Theory201157210311045 – reference: BestM.R.Perfect codes hardly existIEEE Trans. Inf. Theory1983293349351 – reference: BiggsN.Perfect codes in graphsJ. Comb. Theory B1973153289296 – reference: EtzionT.Configuration distribution and designs of codes in the Johnson schemeJ. Comb. Des.20071511534 – reference: Aigner M.: Combinatorial Theory. Springer, New York (1979). – reference: DelsarteP.An algebraic approach to association schemes and coding theoryPhilips J. Res.197310197 – reference: MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. North-Holland, Amsterdam (1977). – reference: ShimabukuroO.On the nonexistence of perfect codes in $$ J(2w + p2, w)$$ J ( 2 w + p 2 , w )Ars Comb.200575129134 – reference: TietäväinenA.On the nonexistence of perfect codes over finite fieldsSIAM J. Appl. Math.19732418896 – reference: ŠpacapanS.Non-existence of face-to-face four dimensional tiling in the Lee metricEur. J. Comb.2007281127133 – reference: BoursP.A.H.On the construction of perfect deletion-correcting codes using design theoryDes. Codes Cryptogr.199561520 – reference: Bertsekas D.P., Gallager R.: Data Networks, 2nd edn. Prentice Hall, Englewood Cliffs (1992). – reference: Kovačević M., Vukobratović D.: Multiset codes for permutation channels. Available online at: arXiv:1301.7564. – reference: KovačevićM.VukobratovićD.Subset codes for packet networksIEEE Commun. Lett.2013174729732 – reference: HorakP.Tilings in Lee metricEur. J. Comb.2009302480489 – reference: HorakP.On perfect Lee codesDiscret. Math.20093091855515561 – reference: GolombS.W.WelchL.R.Perfect codes in the Lee metric and the packing of polyominoesSIAM J. Appl. Math.1970182302317 – reference: ZinovievV.A.LeontievV.K.The nonexistence of perfect codes over Galois fieldsProbl. Control Inf. Theory19732123132 – reference: van Lint J. H.: Nonexistence theorems for perfect error-correcting codes. In: Computers in Algebra and Number Theory, vol. IV, SIAM-AMS Proceedings (1971). – reference: AstolaJ.On perfect Lee codes over small alphabets of odd cardinalityDiscret. Appl. Math.19824227228 – reference: Bange D.W., Barkauskas A.E., Slater P.J.: Efficient dominating sets in graphs. In: Ringeisen R.D., Roberts F.S. (eds.) Applications of Discrete Mathematics, pp. 189–199. SIAM, Philadelphia (1988). – reference: MartinW.J.ZhuX.J.Anticodes for the Grassmann and bilinear forms graphsDes. Codes Cryptogr.1995617379 – reference: EtzionT.On the nonexistence of perfect codes in the Johnson schemeSIAM J. Discret. Math.199692201209 – reference: EtzionT.VardyA.Error-correcting codes in projective spaceIEEE Trans. Inf. Theory201157211651173 – reference: LevenshteinV.I.On perfect codes in deletion and insertion metricDiscret. Math. Appl.199223241258 – reference: Cohen G., Honkala I., Litsyn S., Lobstein A.: Covering Codes. Elsevier, Amsterdam (1997). – reference: PostK.A.Nonexistence theorem on perfect Lee codes over large alphabetsInf. Control1975294369380 – ident: 9893_CR12 – ident: 9893_CR14 – ident: 9893_CR35 – ident: 9893_CR8 doi: 10.1007/BF01390767 – ident: 9893_CR2 doi: 10.1007/s10623-009-9273-3 – ident: 9893_CR20 – ident: 9893_CR7 – ident: 9893_CR9 – ident: 9893_CR22 – ident: 9893_CR5 – ident: 9893_CR23 – ident: 9893_CR25 – ident: 9893_CR27 – ident: 9893_CR1 doi: 10.1007/978-1-4615-6666-3 – ident: 9893_CR19 – ident: 9893_CR17 – ident: 9893_CR30 doi: 10.1016/S0019-9958(75)80005-2 – ident: 9893_CR32 – ident: 9893_CR11 – ident: 9893_CR13 – ident: 9893_CR36 – ident: 9893_CR15 – ident: 9893_CR34 – ident: 9893_CR29 doi: 10.1007/BF01390772 – ident: 9893_CR6 – ident: 9893_CR21 – ident: 9893_CR4 – ident: 9893_CR3 – ident: 9893_CR26 – ident: 9893_CR24 – ident: 9893_CR28 – ident: 9893_CR18 – ident: 9893_CR31 – ident: 9893_CR16 – ident: 9893_CR33 – ident: 9893_CR10 |
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| Snippet | We study the problem of existence of (nontrivial) perfect codes in the discrete
n
-simplex
Δ
ℓ
n
:
=
x
0
,
…
,
x
n
:
x
i
∈
Z
+
,
∑
i
x
i
=
ℓ
under
ℓ
1
metric.... |
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| SubjectTerms | Circuits Coding and Information Theory Computer Science Cryptology Data Structures and Information Theory Discrete Mathematics in Computer Science Information and Communication |
| Title | Perfect codes in the discrete simplex |
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