Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes
Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a pro...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Özbudak, Ferruh [verfasserIn] |
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| Format: |
E-Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2014transfer abstract |
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| Schlagwörter: |
Simultaneous polynomial reconstruction |
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| Umfang: |
13 |
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| Übergeordnetes Werk: |
Enthalten in: Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries - Schweiss, Rüdiger ELSEVIER, 2015transfer abstract, the journal of the EATCS, Amsterdam [u.a.] |
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| Übergeordnetes Werk: |
volume:520 ; year:2014 ; day:6 ; month:02 ; pages:111-123 ; extent:13 |
| Links: |
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| DOI / URN: |
10.1016/j.tcs.2013.10.025 |
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ELV039336867 |
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| 520 | |a Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. | ||
| 520 | |a Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. | ||
| 650 | 7 | |a Simultaneous polynomial reconstruction |2 Elsevier | |
| 650 | 7 | |a Interleaved Reed–Solomon codes |2 Elsevier | |
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10.1016/j.tcs.2013.10.025 doi GBVA2014010000025.pica (DE-627)ELV039336867 (ELSEVIER)S0304-3975(13)00805-0 DE-627 ger DE-627 rakwb eng 004 004 DE-600 620 VZ 690 VZ 50.92 bkl Özbudak, Ferruh verfasserin aut Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes 2014transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Simultaneous polynomial reconstruction Elsevier Interleaved Reed–Solomon codes Elsevier Reed–Solomon codes Elsevier Folded Hermitian codes Elsevier Yayla, Oğuz oth Enthalten in Elsevier Schweiss, Rüdiger ELSEVIER Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries 2015transfer abstract the journal of the EATCS Amsterdam [u.a.] (DE-627)ELV013125583 volume:520 year:2014 day:6 month:02 pages:111-123 extent:13 https://doi.org/10.1016/j.tcs.2013.10.025 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_22 GBV_ILN_40 50.92 Meerestechnik VZ AR 520 2014 6 0206 111-123 13 045F 004 |
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10.1016/j.tcs.2013.10.025 doi GBVA2014010000025.pica (DE-627)ELV039336867 (ELSEVIER)S0304-3975(13)00805-0 DE-627 ger DE-627 rakwb eng 004 004 DE-600 620 VZ 690 VZ 50.92 bkl Özbudak, Ferruh verfasserin aut Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes 2014transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Simultaneous polynomial reconstruction Elsevier Interleaved Reed–Solomon codes Elsevier Reed–Solomon codes Elsevier Folded Hermitian codes Elsevier Yayla, Oğuz oth Enthalten in Elsevier Schweiss, Rüdiger ELSEVIER Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries 2015transfer abstract the journal of the EATCS Amsterdam [u.a.] (DE-627)ELV013125583 volume:520 year:2014 day:6 month:02 pages:111-123 extent:13 https://doi.org/10.1016/j.tcs.2013.10.025 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_22 GBV_ILN_40 50.92 Meerestechnik VZ AR 520 2014 6 0206 111-123 13 045F 004 |
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10.1016/j.tcs.2013.10.025 doi GBVA2014010000025.pica (DE-627)ELV039336867 (ELSEVIER)S0304-3975(13)00805-0 DE-627 ger DE-627 rakwb eng 004 004 DE-600 620 VZ 690 VZ 50.92 bkl Özbudak, Ferruh verfasserin aut Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes 2014transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Simultaneous polynomial reconstruction Elsevier Interleaved Reed–Solomon codes Elsevier Reed–Solomon codes Elsevier Folded Hermitian codes Elsevier Yayla, Oğuz oth Enthalten in Elsevier Schweiss, Rüdiger ELSEVIER Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries 2015transfer abstract the journal of the EATCS Amsterdam [u.a.] (DE-627)ELV013125583 volume:520 year:2014 day:6 month:02 pages:111-123 extent:13 https://doi.org/10.1016/j.tcs.2013.10.025 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_22 GBV_ILN_40 50.92 Meerestechnik VZ AR 520 2014 6 0206 111-123 13 045F 004 |
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10.1016/j.tcs.2013.10.025 doi GBVA2014010000025.pica (DE-627)ELV039336867 (ELSEVIER)S0304-3975(13)00805-0 DE-627 ger DE-627 rakwb eng 004 004 DE-600 620 VZ 690 VZ 50.92 bkl Özbudak, Ferruh verfasserin aut Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes 2014transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Simultaneous polynomial reconstruction Elsevier Interleaved Reed–Solomon codes Elsevier Reed–Solomon codes Elsevier Folded Hermitian codes Elsevier Yayla, Oğuz oth Enthalten in Elsevier Schweiss, Rüdiger ELSEVIER Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries 2015transfer abstract the journal of the EATCS Amsterdam [u.a.] (DE-627)ELV013125583 volume:520 year:2014 day:6 month:02 pages:111-123 extent:13 https://doi.org/10.1016/j.tcs.2013.10.025 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_22 GBV_ILN_40 50.92 Meerestechnik VZ AR 520 2014 6 0206 111-123 13 045F 004 |
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10.1016/j.tcs.2013.10.025 doi GBVA2014010000025.pica (DE-627)ELV039336867 (ELSEVIER)S0304-3975(13)00805-0 DE-627 ger DE-627 rakwb eng 004 004 DE-600 620 VZ 690 VZ 50.92 bkl Özbudak, Ferruh verfasserin aut Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes 2014transfer abstract 13 nicht spezifiziert zzz rdacontent nicht spezifiziert z rdamedia nicht spezifiziert zu rdacarrier Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. Simultaneous polynomial reconstruction Elsevier Interleaved Reed–Solomon codes Elsevier Reed–Solomon codes Elsevier Folded Hermitian codes Elsevier Yayla, Oğuz oth Enthalten in Elsevier Schweiss, Rüdiger ELSEVIER Influence of bulk fibre properties of PAN-based carbon felts on their performance in vanadium redox flow batteries 2015transfer abstract the journal of the EATCS Amsterdam [u.a.] (DE-627)ELV013125583 volume:520 year:2014 day:6 month:02 pages:111-123 extent:13 https://doi.org/10.1016/j.tcs.2013.10.025 Volltext GBV_USEFLAG_U GBV_ELV SYSFLAG_U GBV_ILN_22 GBV_ILN_40 50.92 Meerestechnik VZ AR 520 2014 6 0206 111-123 13 045F 004 |
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improved probabilistic decoding of interleaved reed–solomon codes and folded hermitian codes |
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Improved probabilistic decoding of interleaved Reed–Solomon codes and folded Hermitian codes |
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Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. |
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Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. |
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Probabilistic simultaneous polynomial reconstruction algorithm of Bleichenbacher, Kiayias, and Yung is extended to the polynomials whose degrees are allowed to be distinct. Specifically, for a finite field F , positive integers n, r, t and distinct elements z 1 , z 2 , … , z n ∈ F , we present a probabilistic algorithm which can recover polynomials p 1 , p 2 , … , p r ∈ F [ x ] of degree less than k 1 , k 2 , … , k r respectively for a given instance 〈 y i , 1 , … , y i , r 〉 i = 1 n satisfying p l ( z i ) = y i , l for all l ∈ { 1 , 2 , … , r } and for all i ∈ I ⊂ { 1 , 2 , … , n } such that | I | = t with probability at least 1 − n − t | F | and with time complexity at most O ( r n 4 ) if t ⩾ max { k 1 , k 2 , … , k r , n + ∑ j = 1 r k j r + 1 } . Next, by using this algorithm, we present a probabilistic decoder for interleaved Reed–Solomon codes. It is observed that interleaved Reed–Solomon codes over F with rate R can be decoded up to burst error rate r r + 1 ( 1 − R ) probabilistically for an interleaving parameter r. Then, it is proved that q-folded Hermitian codes over F q 2 q with rate R can be decoded up to error rate q q + 1 ( 1 − R ) probabilistically. |
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