### Citations

2632 | Theory of Error Correcting Codes - MacWilliams, Sloane - 1977 |

2483 |
Quantum computation and quantum information
- Nielsen, Chuang
- 2000
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Citation Context ...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope... |

303 | Quantum error correction via codes over GF(4
- Calderbank, Rains, et al.
- 1998
(Show Context)
Citation Context ...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope... |

283 |
Fundamentals of error-correcting codes
- Huffman, Pless
- 2003
(Show Context)
Citation Context ... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po... |

81 | Nonbinary Quantum Stabilizer Codes - Ashikhmin, Knill |

63 | On behaviors and convolutional codes, - Rosenthal, Schumacher, et al. - 1996 |

52 |
Convolutional Codes: An Algebraic Approach.
- Piret
- 1988
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Citation Context ...In this paper, we utilize the class of negacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convolutional codes. More precisely, we apply the famous method proposed by Piret =-=[39]-=- (generalized recently by Aly et al. [2]), which consists in the construction of (classical) convolutional codes derived from block codes. An advantage of our techniques of construction lie in the fac... |

49 | Nonbinary stabilizer codes over finite fields
- Ketkar, Klappenecker, et al.
(Show Context)
Citation Context ...uctions presented in this paper are performed algebraically and not by computational search. 1 Introduction Much effort have been paid in order to construct good quantum error-correcting codes (QECC) =-=[4, 9, 22, 24, 25, 36, 44]-=- as well as quantum convolutional codes with good parameters [1–3, 13–16,27, 37, 38]. On the other hand, the investigation of the class of (classical) convolutional codes and their corresponding prope... |

40 | Maximum distance separable convolutional codes. Applicable Algebra in Eng
- Rosenthal, Smarandache
- 1998
(Show Context)
Citation Context ...assical) convolutional codes and their corresponding properties as well as constructions of maximum-distance-separable (MDS) convolutional codes (i.e., codes attaining the generalized Singleton bound =-=[41]-=-) have also been presented in the literature [11, 12, 17, 26–34,39, 41–43]. In this paper, we utilize the class of negacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convol... |

34 |
Convolutional codes I: Algebraic structure
- Jr
- 1970
(Show Context)
Citation Context ... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po... |

34 | Enlargement of Calderbank-Shor-Steane quantum codes,”
- Steane
- 1999
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Citation Context |

23 | Convolutional and tail-biting quantum error-correcting codes,”
- Forney, Grassl, et al.
- 2007
(Show Context)
Citation Context ...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To... |

22 | Strongly MDS convolutional codes - Gluesing-Luerssen, Rosenthal, et al. |

21 | Constructions of MDSConvolutional Codes,
- Smarandache, Luerssen, et al.
- 2001
(Show Context)
Citation Context ...more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or minimal =-=[18, 32, 43]-=-) if the overall constraint length γ = k∑ i=1 γi, where γi = max1≤j≤n{deg gij}, has the smallest value among all basic generator matrices (in this case the overall constraint length γ will be called t... |

19 | Convolutional codes with maximum distance profile - Hutchinson, Rosenthal, et al. - 2005 |

18 | Non-catastrophic encoders and encoder inverses for quantum convolutional codes - Grassl, Rötteler - 2006 |

12 | Short unit-memory byte-oriented binary convolutional codes having maximal free distance,” - Lee - 1976 |

12 | Description of a quantum convolutional code,” - Ollivier, Tillich - 2003 |

9 | Distance bounds for convolutional codes and some optimal codes. e-print arXiv:math/0305135 - Gluesing-Luerssen, Schmale |

9 | A convolutional equivalent to Reed-Solomon codes - Piret - 1988 |

8 |
The structure of 1-generator quasi-twisted codes and new linear codes
- Aydin, Siap, et al.
- 2001
(Show Context)
Citation Context ...n). The minimal polynomial (over Fq2) of β j ∈ Fq2m is denoted byM (j)(x) and it is given byM (j)(x) = ∏ j∈Ci (x−βj). The dimension of C is given by n − |Z|. The BCH bound for Constacyclic codes (see =-=[5, 23]-=-) asserts that is C is a q2-ary negacyclic code of length n with 2 generator polynomial g(x) and if g(x) has the elements {β2i+1|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth root of unity, the... |

8 |
Constructions of new families of nonbinary quantum codes, Phys
- Guardia
- 2009
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Citation Context |

6 | Negacyclic Codes for the Lee Metric - Berlekamp - 1968 |

6 | Quantum block and convolutional codes from self-orthogonal product codes,” in
- Grassl, Rotteler
- 2005
(Show Context)
Citation Context ...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To... |

6 | On classes of convolutional codes that are not asymptotically catastrophic - Hole - 2000 |

6 |
New quantum MDS codes
- Guardia
- 2011
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5 | Quantum convolutional codes derived from Reed-Solomon and Reed-Muller codes
- Aly, Klappenecker, et al.
- 2007
(Show Context)
Citation Context ... β is a primitive 2nth root of unity, then the minimum distance dC of C satisfies dC ≥ d. 3 Classical Convolutional Codes The class of (classical) convolutional codes is a well-studied class of codes =-=[2, 3, 12, 18, 19, 39]-=-. We assume the reader is familiar with the theory of convolutional codes. For more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a po... |

5 | A matrix ring description for cyclic convolutional codes
- Gluesing-Luerssen, Tsang
(Show Context)
Citation Context ...more details, see [19]. Recall that a polynomial encoder matrix G(D) = (gij) ∈ Fq[D] k×n is called basic if G(D) has a polynomial right inverse. A basic generator matrix is called reduced (or minimal =-=[18, 32, 43]-=-) if the overall constraint length γ = k∑ i=1 γi, where γi = max1≤j≤n{deg gij}, has the smallest value among all basic generator matrices (in this case the overall constraint length γ will be called t... |

4 | Jr.. A concatenated [(4 - Almeida, Palazzo - 2004 |

4 |
Quantum convolutional BCH codes. e-print arXiv:quant-ph/0703113. 9[3
- Aly, Grassl, et al.
(Show Context)
Citation Context ...gacyclic codes [5–8, 10, 23] in order to construct classical and quantum MDS convolutional codes. More precisely, we apply the famous method proposed by Piret [39] (generalized recently by Aly et al. =-=[2]-=-), which consists in the construction of (classical) convolutional codes derived from block codes. An advantage of our techniques of construction lie in the fact that all new (classical and quantum) c... |

4 | Constacyclic codes over finite fields - Chen, Fan, et al. - 2012 |

4 | Linear system modelization of concatenated block and convolutional codes. Linear Algebra and its Applications - Climent, Herranz, et al. - 2008 |

4 |
New quantum MDS codes from negacyclic codes
- Kai, Zhu
- 2013
(Show Context)
Citation Context ... Theorem 6.2 [3] (Quantum Singleton bound) The free distance of an [(n, k, µ; γ, df )]q Fq2-linear pure convolutional stabilizer code is bounded by df ≤ n− k 2 (⌊ 2γ n+ k ⌋ + 1 ) + γ + 1. 8 Lemma 6.3 =-=[20]-=- Let n = q2 + 1, where q ≡ 1 (mod 4) is a power of an odd prime and suppose that s = n/2. If C is a q2-ary negacyclic code of length n with defining set Z = ∪δi=0Cs−2i, where 0 ≤ δ ≤ (q − 1)/2, then C... |

4 |
Pseudocyclic maximum-distance-separable codes
- Krishna, Sarwate
- 1990
(Show Context)
Citation Context ...n). The minimal polynomial (over Fq2) of β j ∈ Fq2m is denoted byM (j)(x) and it is given byM (j)(x) = ∏ j∈Ci (x−βj). The dimension of C is given by n − |Z|. The BCH bound for Constacyclic codes (see =-=[5, 23]-=-) asserts that is C is a q2-ary negacyclic code of length n with 2 generator polynomial g(x) and if g(x) has the elements {β2i+1|0 ≤ i ≤ d − 2} as roots, where β is a primitive 2nth root of unity, the... |

4 | On doubly-cyclic convolutional codes,” Applicable Algebra in Engineering, - Gluesing-Luerssen, Schmale - 2006 |

3 | A class of constacyclic codes over a finite field - Bakshi, Raka |

2 |
Negacyclic duadic codes
- Blackford
(Show Context)
Citation Context ...ribed. 2 Negacyclic codes The class of negacyclic codes [5–8, 10, 21, 23] have been studied in the literature. This class of codes are a particular class of a more general class of constacyclic codes =-=[8]-=-. In this section we review the basic concepts of these codes. Throughout this paper, we always assume that q is a power of an odd prime, Fq is a finite field with q elements and n is a positive integ... |

2 |
Constructions of quantum convolutional codes. e-print arXiv:quant-ph/0703182
- Grassl, Rötteler
(Show Context)
Citation Context ...s 5.5 and 5.6, the result follows. 6 New Quantum MDS-Convolutional codes As in the classical case, the construction of MDS quantum convolutional codes is a difficult task. This task is performed in =-=[3, 13, 14, 16]-=- but only in [3, 14] the constructions are made algebraically. Here, we propose the construction of MDS convolutional stabilizer codes derived from the convolutional codes constructed in Section 5. To... |

2 | Convolutional Codes Derived From Group Character Codes. e-print arXiv:1212.4653. [18 - Guardia |

2 | Quantum convolutional codes: fundamentals. e-print arXiv:quant-ph/0401134 - Ollivier, Tillich |

2 | A class of constacyclic codes over Zpm , Finite Fields - Zhu, Kai |

1 | Quantum Negacyclic Codes - Kai, Zhu |

1 | On classical and quantum MDS-convolutional BCH codes - Guardia |