Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs
We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math>&...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Guruswami, Venkatesan [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Systematik: |
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| Übergeordnetes Werk: |
Enthalten in: IEEE transactions on information theory - Piscataway, NJ : IEEE, 1963, 62(2016), 5, Seite 2707-2718 |
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| Übergeordnetes Werk: |
volume:62 ; year:2016 ; number:5 ; pages:2707-2718 |
| Links: |
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| DOI / URN: |
10.1109/TIT.2016.2544347 |
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| Katalog-ID: |
OLC1974709078 |
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| 100 | 1 | |a Guruswami, Venkatesan |e verfasserin |4 aut | |
| 245 | 1 | 0 | |a Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs |
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| 520 | |a We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. | ||
| 650 | 4 | |a Space-time codes | |
| 650 | 4 | |a Measurement | |
| 650 | 4 | |a Context | |
| 650 | 4 | |a Network coding | |
| 650 | 4 | |a Decoding | |
| 650 | 4 | |a Error analysis | |
| 650 | 4 | |a algebraic codes | |
| 650 | 4 | |a Reed-Solomon codes | |
| 650 | 4 | |a list error-correction | |
| 650 | 4 | |a pseudorandomness | |
| 650 | 4 | |a Linear codes | |
| 650 | 4 | |a Algebra | |
| 650 | 4 | |a Information theory | |
| 650 | 4 | |a Polynomials | |
| 700 | 1 | |a Wang, Carol |4 oth | |
| 700 | 1 | |a Xing, Chaoping |4 oth | |
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10.1109/TIT.2016.2544347 doi PQ20160610 (DE-627)OLC1974709078 (DE-599)GBVOLC1974709078 (PRQ)i826-3186ec3a0bbaf51cc503d68ade2a8e673313d7b6430973aa41947f99b526aa020 (KEY)0023448620160000062000502707explicitlistdecodablerankmetricandsubspacecodesvia DE-627 ger DE-627 rakwb eng 070 620 DNB SA 5570 AVZ rvk Guruswami, Venkatesan verfasserin aut Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. Space-time codes Measurement Context Network coding Decoding Error analysis algebraic codes Reed-Solomon codes list error-correction pseudorandomness Linear codes Algebra Information theory Polynomials Wang, Carol oth Xing, Chaoping oth Enthalten in IEEE transactions on information theory Piscataway, NJ : IEEE, 1963 62(2016), 5, Seite 2707-2718 (DE-627)12954731X (DE-600)218505-2 (DE-576)01499819X 0018-9448 nnns volume:62 year:2016 number:5 pages:2707-2718 http://dx.doi.org/10.1109/TIT.2016.2544347 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-BBI GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2088 SA 5570 AR 62 2016 5 2707-2718 |
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10.1109/TIT.2016.2544347 doi PQ20160610 (DE-627)OLC1974709078 (DE-599)GBVOLC1974709078 (PRQ)i826-3186ec3a0bbaf51cc503d68ade2a8e673313d7b6430973aa41947f99b526aa020 (KEY)0023448620160000062000502707explicitlistdecodablerankmetricandsubspacecodesvia DE-627 ger DE-627 rakwb eng 070 620 DNB SA 5570 AVZ rvk Guruswami, Venkatesan verfasserin aut Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. Space-time codes Measurement Context Network coding Decoding Error analysis algebraic codes Reed-Solomon codes list error-correction pseudorandomness Linear codes Algebra Information theory Polynomials Wang, Carol oth Xing, Chaoping oth Enthalten in IEEE transactions on information theory Piscataway, NJ : IEEE, 1963 62(2016), 5, Seite 2707-2718 (DE-627)12954731X (DE-600)218505-2 (DE-576)01499819X 0018-9448 nnns volume:62 year:2016 number:5 pages:2707-2718 http://dx.doi.org/10.1109/TIT.2016.2544347 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-BBI GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2088 SA 5570 AR 62 2016 5 2707-2718 |
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10.1109/TIT.2016.2544347 doi PQ20160610 (DE-627)OLC1974709078 (DE-599)GBVOLC1974709078 (PRQ)i826-3186ec3a0bbaf51cc503d68ade2a8e673313d7b6430973aa41947f99b526aa020 (KEY)0023448620160000062000502707explicitlistdecodablerankmetricandsubspacecodesvia DE-627 ger DE-627 rakwb eng 070 620 DNB SA 5570 AVZ rvk Guruswami, Venkatesan verfasserin aut Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. Space-time codes Measurement Context Network coding Decoding Error analysis algebraic codes Reed-Solomon codes list error-correction pseudorandomness Linear codes Algebra Information theory Polynomials Wang, Carol oth Xing, Chaoping oth Enthalten in IEEE transactions on information theory Piscataway, NJ : IEEE, 1963 62(2016), 5, Seite 2707-2718 (DE-627)12954731X (DE-600)218505-2 (DE-576)01499819X 0018-9448 nnns volume:62 year:2016 number:5 pages:2707-2718 http://dx.doi.org/10.1109/TIT.2016.2544347 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-BBI GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2088 SA 5570 AR 62 2016 5 2707-2718 |
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10.1109/TIT.2016.2544347 doi PQ20160610 (DE-627)OLC1974709078 (DE-599)GBVOLC1974709078 (PRQ)i826-3186ec3a0bbaf51cc503d68ade2a8e673313d7b6430973aa41947f99b526aa020 (KEY)0023448620160000062000502707explicitlistdecodablerankmetricandsubspacecodesvia DE-627 ger DE-627 rakwb eng 070 620 DNB SA 5570 AVZ rvk Guruswami, Venkatesan verfasserin aut Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. Space-time codes Measurement Context Network coding Decoding Error analysis algebraic codes Reed-Solomon codes list error-correction pseudorandomness Linear codes Algebra Information theory Polynomials Wang, Carol oth Xing, Chaoping oth Enthalten in IEEE transactions on information theory Piscataway, NJ : IEEE, 1963 62(2016), 5, Seite 2707-2718 (DE-627)12954731X (DE-600)218505-2 (DE-576)01499819X 0018-9448 nnns volume:62 year:2016 number:5 pages:2707-2718 http://dx.doi.org/10.1109/TIT.2016.2544347 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-BBI GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2088 SA 5570 AR 62 2016 5 2707-2718 |
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10.1109/TIT.2016.2544347 doi PQ20160610 (DE-627)OLC1974709078 (DE-599)GBVOLC1974709078 (PRQ)i826-3186ec3a0bbaf51cc503d68ade2a8e673313d7b6430973aa41947f99b526aa020 (KEY)0023448620160000062000502707explicitlistdecodablerankmetricandsubspacecodesvia DE-627 ger DE-627 rakwb eng 070 620 DNB SA 5570 AVZ rvk Guruswami, Venkatesan verfasserin aut Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. Space-time codes Measurement Context Network coding Decoding Error analysis algebraic codes Reed-Solomon codes list error-correction pseudorandomness Linear codes Algebra Information theory Polynomials Wang, Carol oth Xing, Chaoping oth Enthalten in IEEE transactions on information theory Piscataway, NJ : IEEE, 1963 62(2016), 5, Seite 2707-2718 (DE-627)12954731X (DE-600)218505-2 (DE-576)01499819X 0018-9448 nnns volume:62 year:2016 number:5 pages:2707-2718 http://dx.doi.org/10.1109/TIT.2016.2544347 Volltext http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470 GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-MAT SSG-OLC-BUB SSG-OPC-BBI GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2088 SA 5570 AR 62 2016 5 2707-2718 |
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Guruswami, Venkatesan |
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Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs |
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explicit list-decodable rank-metric and subspace codes via subspace designs |
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Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs |
| abstract |
We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. |
| abstractGer |
We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. |
| abstract_unstemmed |
We construct an explicit family of <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-linear rank-metric codes over any field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula> that enables efficient list-decoding up to a fraction <inline-formula> <tex-math notation="LaTeX">\rho </tex-math></inline-formula> of errors in the rank metric with a rate of <inline-formula> <tex-math notation="LaTeX">1-\rho - \varepsilon </tex-math></inline-formula>, for any desired <inline-formula> <tex-math notation="LaTeX">\rho \in (0,1) </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX"> \varepsilon > 0 </tex-math></inline-formula>. This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes. |
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Explicit List-Decodable Rank-Metric and Subspace Codes via Subspace Designs |
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This is the first explicit construction of positive rate rank-metric codes for efficient list-decoding beyond the unique decoding radius. Our codes are explicit subcodes of the well-known Gabidulin codes, which encode linearized polynomials of low degree via their values at a collection of linearly independent points. The subcode is picked by restricting the message polynomials to an <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h} </tex-math></inline-formula>-subspace that evades the structured subspaces over an extension field <inline-formula> <tex-math notation="LaTeX"> \mathbb {F}_{h^{t}} </tex-math></inline-formula> that arise in our linear-algebraic list decoder for Gabidulin codes. This subspace is obtained by combining subspace designs constructed by Guruswami and Kopparty (FOCS'13) with subspace-evasive varieties due to Dvir and Lovett (STOC'12). We establish a similar result for subspace codes, which have received much attention recently in the context of network coding. We also give explicit subcodes of folded Reed-Solomon (RS) codes with small folding order, which are list-decodable (in the Hamming metric) with optimal redundancy, motivated by the fact that list-decoding RS codes reduces to list-decoding such folded RS codes. However, as we only list-decode a subcode of these codes, the Johnson radius continues to be the best known error fraction for list-decoding RS codes.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Space-time codes</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Measurement</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Context</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Network coding</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Decoding</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Error analysis</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">algebraic codes</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Reed-Solomon codes</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">list error-correction</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">pseudorandomness</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Linear codes</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Algebra</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Information theory</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Polynomials</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Wang, Carol</subfield><subfield code="4">oth</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Xing, Chaoping</subfield><subfield code="4">oth</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">IEEE transactions on information theory</subfield><subfield code="d">Piscataway, NJ : IEEE, 1963</subfield><subfield code="g">62(2016), 5, Seite 2707-2718</subfield><subfield code="w">(DE-627)12954731X</subfield><subfield code="w">(DE-600)218505-2</subfield><subfield code="w">(DE-576)01499819X</subfield><subfield code="x">0018-9448</subfield><subfield code="7">nnns</subfield></datafield><datafield tag="773" ind1="1" ind2="8"><subfield code="g">volume:62</subfield><subfield code="g">year:2016</subfield><subfield code="g">number:5</subfield><subfield code="g">pages:2707-2718</subfield></datafield><datafield tag="856" ind1="4" ind2="1"><subfield code="u">http://dx.doi.org/10.1109/TIT.2016.2544347</subfield><subfield code="3">Volltext</subfield></datafield><datafield tag="856" ind1="4" ind2="2"><subfield code="u">http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=7437470</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_USEFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SYSFLAG_A</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_OLC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-TEC</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-MAT</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OLC-BUB</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">SSG-OPC-BBI</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_65</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_70</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_2002</subfield></datafield><datafield tag="912" ind1=" " ind2=" "><subfield code="a">GBV_ILN_2088</subfield></datafield><datafield tag="936" ind1="r" ind2="v"><subfield code="a">SA 5570</subfield></datafield><datafield tag="951" ind1=" " ind2=" "><subfield code="a">AR</subfield></datafield><datafield tag="952" ind1=" " ind2=" "><subfield code="d">62</subfield><subfield code="j">2016</subfield><subfield code="e">5</subfield><subfield code="h">2707-2718</subfield></datafield></record></collection>
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