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Subfield-subcodes of Generalized Toric codes

Fernando Hernando, Michael E. O'Sullivan, Emanuel Popovici, Shraddha Srivastava
2010 2010 IEEE International Symposium on Information Theory  
We study subfield-subcodes of Generalized Toric (GT) codes over F_p^s. These are the multidimensional analogues of BCH codes, which may be seen as subfield-subcodes of generalized Reed-Solomon codes.  ...  We identify polynomial generators for subfield-subcodes of GT codes which allows us to determine the dimensions and obtain bounds for the minimum distance.  ...  Notice that p = 2 ∤ s = 3 thus from Theorem 4.1 we know that the dual of a subfield-subcode is again the subfield-subcode of another toric code.  ... 
doi:10.1109/isit.2010.5513688 dblp:conf/isit/HernandoOPS10 fatcat:72kzmgcr3vd2ljiqgtn4btimie

The Dimension of Subcode-Subfields of Shortened Generalized Reed Solomon Codes [article]

Fernando Hernando and Kyle Marshall and Michael E. O'Sullivan
2011 arXiv   pre-print
Generalized Reed-Solomon (GRS) codes are a generalization of RS codes, whose subfield-subcodes are extensively studied.  ...  In this paper we show that a particular class of GRS codes produces many subfield-subcodes with large dimension.  ...  Closed formulae are obtained for the true dimension of certain alternant codes in [4] , for subfield-subcodes of toric codes in [8] and for a family of Goppa codes in [14] .  ... 
arXiv:1108.5475v1 fatcat:bz5tyuw5vfeufowyw6nlebl5xu

New binary and ternary LCD codes [article]

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2017 arXiv   pre-print
Subfield-subcodes of J-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes.  ...  We describe binary and ternary LCD codes constructed as subfield-subcodes of J-affine variety codes and provide some new and good LCD codes coming from this construction.  ...  LCD codes; complementary dual; subfield subcodes; J-affine variety codes; toric codes.  ... 
arXiv:1710.00196v2 fatcat:pir252tsbvblfcyves63snvayi

On Counting Subring-Subcodes of Free Linear Codes Over Finite Principal Ideal Rings [article]

Ramakrishna Bandi and Alexandre Fotue Tabue and Edgar Martínez-Moro
2016 arXiv   pre-print
For k and k_0, positive integers we determine the number of free S-linear codes B of length l with the property k = rank_S(B) and k_0 = rank_R (B∩ R^l).  ...  Let R be a finite principal ideal ring and S the Galois extension of R of degree m.  ...  Introduction Many codes over a finite field F can be seen as subfield-subcodes of codes that are defined over a some extension field of F.  ... 
arXiv:1612.02213v1 fatcat:3hmj4l2bsfhfdcc5ltkvhg3xsa

Quantum codes from affine variety codes and their subfield-subcodes [article]

Carlos Galindo, Fernando Hernando
2014 arXiv   pre-print
We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction.  ...  With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.  ...  Quantum codes from subfield-subcodes of affine variety codes We devote this section to explain which of our affine variety subfield-subcodes yield stabilizer codes.  ... 
arXiv:1403.4060v2 fatcat:koyl7hargzcclkuustlb663vqy

Quantum codes from affine variety codes and their subfield-subcodes

Carlos Galindo, Fernando Hernando
2014 Designs, Codes and Cryptography  
We use affine variety codes and their subfield-subcodes for obtaining quantum stabilizer codes via the CSS code construction.  ...  With this procedure, we get codes with good parameters and a code whose parameters exceed the CSS quantum Gilbert-Varshamov bound given by Feng and Ma.  ...  Quantum codes from subfield-subcodes of affine variety codes We devote this section to explain which of our affine variety subfield-subcodes yield stabilizer codes.  ... 
doi:10.1007/s10623-014-0016-8 fatcat:lkpxzvnfqbhsfproy5hdafnsbu

New binary and ternary LCD codes

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2018 IEEE Transactions on Information Theory  
Subfield-subcodes of J-affine variety codes are a generalization of BCH codes which have been successfully used for constructing good quantum codes.  ...  Hence, the study of LCD codes is mainly open for binary and ternary fields.  ...  Binary and ternary LCD subfield-subcodes coming from the univariate case We devote this section to provide binary and ternary LCD codes obtained as subfield-subcodes of univariate J-affine variety codes  ... 
doi:10.1109/tit.2018.2834500 fatcat:je7emek3zbbihcffeln6bj24nu

Weighted Reed–Muller codes revisited

Olav Geil, Casper Thomsen
2012 Designs, Codes and Cryptography  
With a small modification one of these algorithms is able to correct up to 31 errors of the [49, 11, 28] Joyner code.  ...  For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms.  ...  The above observation guarantees that codes E(M, S) in general can be seen as subcodes of subfield subcodes of certain Reed-Solomon codes over F q m , but it is not straightforward which elements of F  ... 
doi:10.1007/s10623-012-9680-8 fatcat:atvj22ye7rgr3jgxwqiaqbz6xm

Weighted Reed-Muller codes revisited [article]

Olav Geil, Casper Thomsen
2011 arXiv   pre-print
With a small modification one of these algorithms is able to correct up to 31 errors of the [49, 11, 28] Joyner code.  ...  For a class of affine variety codes that contains the weighted Reed-Muller codes we then present two list decoding algorithms.  ...  The above observation guarantees that codes E(M, S) in general can be seen as subcodes of subfield subcodes of certain Reed-Solomon codes over F q m , but it is not straightforward which elements of F  ... 
arXiv:1108.6185v1 fatcat:fw5ylmg7wnbtxmnr3wmuwmgffe

List decoding of a class of affine variety codes [article]

Olav Geil, Casper Thomsen
2011 arXiv   pre-print
We then use this information to design a list decoding algorithm for a large class of affine variety codes.  ...  When given the leading monomial of F with respect to a lexicographic ordering we derive improved information on the possible number of zeros of F of multiplicity at least r from S.  ...  It is well-known that the generalized Reed-Muller codes can be viewed as subfield subcodes of Reed-Solomon codes.  ... 
arXiv:1101.4999v1 fatcat:tpnnl5qxwnf6vmfbiarirlbrgi

Algebraic geometry codes from higher dimensional varieties [article]

John B. Little
2008 arXiv   pre-print
Connections with the theories of toric codes and order domains are also briefly indicated.  ...  This paper is a general survey of literature on Goppa-type codes from higher dimensional algebraic varieties.  ...  Some ideas about finding good subcodes of the C 2 codes have been presented by Brouwer in [4] .  ... 
arXiv:0802.2349v1 fatcat:dmofl7jusvbexbagczm3jpetou

Improved Constructions of Nested Code Pairs

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2018 IEEE Transactions on Information Theory  
Keywords: asymmetric quantum code, CSS construction, Feng-Rao bound, nested codes, ramp secret sharing, relative generalized Hamming weight, relative minimum distance, wiretap channel of type II.  ...  Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes [40] .  ...  Another advantage of J-affine variety codes is that they are suited for the construction of subfield subcodes.  ... 
doi:10.1109/tit.2017.2755682 fatcat:xlf45hnajrbhblqnidv57p4thm

Improved constructions of nested code pairs [article]

Carlos Galindo, Olav Geil, Fernando Hernando, Diego Ruano
2017 arXiv   pre-print
Such pairs of nested codes are indispensable for the determination of good linear ramp secret sharing schemes [35].  ...  By this we mean that for any two out of the three parameters the third parameter of the constructed code pair is large.  ...  Another advantage of J-affine variety codes is that they are suited for the construction of subfield subcodes.  ... 
arXiv:1610.06363v3 fatcat:pnlsoiwrlncdldzen3scubzyhy

Mitigating Coherent Noise by Balancing Weight-2 Z-Stabilizers [article]

Jingzhen Hu, Qingzhong Liang, Narayanan Rengaswamy, Robert Calderbank
2021 arXiv   pre-print
Further, the weight-2 Z-stabilizers generate a direct product of single-parity-check codes with even block length.  ...  The conditions for a stabilizer code being preserved by transversal 2π/2^l Z-rotations at 4 ≤ l ≤ l_max <∞ level in the Clifford hierarchy lead to generalizations of Gleason's theorem that may be of independent  ...  Any opinions, findings, conclusions, and recommendations expressed in this material are those of the authors and do not necessarily reflect the views of these sponsors.  ... 
arXiv:2011.00197v5 fatcat:yy5t6hdl65h3zeleunpjt2hw4e

Subject index volumes 1–200

1999 Theoretical Computer Science  
the context-free language family, 37 subfield subcodes of Goppa codes, 1753 subformula property, 3 182 subframe Iogics in Walter's sense, 3693 subgrammar, 3808 subgraph, 2490,2706,2772 biconnected  ...  arrays into -, 232 families of -, 2175, 2246 feature -, 2336, 2749 finite symbolic -, 2632 fundamental properties of infinite -, 511 generating -, 605 grammatical codes of -, 2394 guingois -, 2244  ... 
doi:10.1016/s0304-3975(98)00319-3 fatcat:s22ud3iiqjht7lfbtc3zctk7zm