#### DMCA

## Kerdock codes and extremal Euclidean line-sets (802)

### Citations

375 |
An algebraic approach to the association schemes of coding theory
- Delsarte
- 1973
(Show Context)
Citation Context ...4 4 −1 √ √ N N 2 − 2 −1 1 −N−√N 2 −N+√N 2 N − 1 ⎛ (N−2)(N−1) N 1 N − 1 2 2 − 1 1 − Q = ⎜ ⎝ √ √ √ N − 1 √ N + 1 −1 1 N − 1 − N + 1 −1 1 −1 −N ⎞ ⎟ ⎠ N 2 + 1 2 − 1 . ⎟ ⎠ , 7Proof. See Delsarte’s thesis =-=[12]-=-, Example 2 on page 82. □ In his thesis [12, Example 2 on page 82] Delsarte considers a binary code Y of length N − 1 with three nonzero distances 2 m ±2 (m−1)/2 , 2 m . He shows that |Y | ≤ N 2 /2, s... |

174 | The Z4-linearity of Kerdock, Preparata, Goethals, and related codes
- Hammons, Kumar, et al.
- 1994
(Show Context)
Citation Context ...heorem 1). The situation is similar to one, when the formal duality between binary nonlinear Kerdock and Preparata codes was explained by duality between quaternary linear Kerdock and Preparata codes =-=[17]-=-. We also note that maximal real mutually unbiased bases determine a 4-class association scheme of size N2 + 2N in RN , as well as a 3-class association scheme of size N2 /2 in RN−1 (which corresponds... |

84 | Z4-Kerdock codes, orthogonal spreads, and extremal Euclidean line-sets
- Calderbank, Cameron, et al.
- 1997
(Show Context)
Citation Context ...attice with automorphism group 2 14 : (2 3 : L3(2)). The lattice ∗ Supported by Japan Society for the Promotion of Science ⎞ ⎟ ⎠ . 1can be obtained by construction A from binary shortened projective =-=[14,4,7]-=- code. Theta series of the lattice is equal to θ(q) = 1 + 28q4 + 1024q7 + 2156q8 + · · ·. So the 64 vectors are not minimal vectors of the lattice. We show that the scheme has a generalization in term... |

64 |
Spherical codes and designs, Geom. Dedicata (6
- Delsarte, Goethals, et al.
- 1977
(Show Context)
Citation Context ...attice with automorphism group 2 14 : (2 3 : L3(2)). The lattice ∗ Supported by Japan Society for the Promotion of Science ⎞ ⎟ ⎠ . 1can be obtained by construction A from binary shortened projective =-=[14,4,7]-=- code. Theta series of the lattice is equal to θ(q) = 1 + 28q4 + 1024q7 + 2156q8 + · · ·. So the 64 vectors are not minimal vectors of the lattice. We show that the scheme has a generalization in term... |

61 | Universally optimal distribution of points on spheres
- Cohn, Kumar
(Show Context)
Citation Context ...and m odd, from Kerdock codes. These association schemes are generalizations of the association scheme of 64 points in R 14 which is a candidate of universal optimal code considered by Cohn and Kumar =-=[9]-=-. In this section we will show that from such Kerdock codes we can obtain the specific line systems in real Euclidean space R N , or equivalently, maximal real MUB (mutually unbiased bases) in real Eu... |

42 | translation planes and Kerdock sets
- Spreads
- 1982
(Show Context)
Citation Context ...construct a 3-class association scheme of size N2 /4, which leads to a spherical code in RN−2 of size N2 /4. As binary Kerdock code we understand a binary code of length N obtained from a Kerdock set =-=[18]-=-, so Kerdock code is inside of a second order Reed-Muller code (see for details Section 3), and it has nonzero distances 2m ± 2 (m−1)/2, 2m , 2m+1 . Furthermore, we define Kerdock-like code as a binar... |

20 |
Bounds for systems of lines, and Jacobi polynomials, Philips Res. Rep
- Delsarte, Goethals, et al.
- 1975
(Show Context)
Citation Context ...idel [14], and does not depend on the specific structures of the codes. Two orthonormal bases B and B ′ in R N are called mutually unbiased if |(x,y)| = 1/ √ N for any x ∈ B and y ∈ B ′ . It is known =-=[7, 13]-=- that there can be at most N/2 + 1 mutually unbiased bases in dimension N, and constructions reaching this upper bound are known only for values N = 2 m+1 . Let us define Kerdock-like codes as binary ... |

18 | Experimental study of energy-minimizing point configurations on spheres, Experiment - Ballinger, Blekherman, et al. |

10 |
Association schemes related to Kasami codes and Kerdock sets
- Caen, Dam
- 1999
(Show Context)
Citation Context ...alently, any Kerdock-like code) determines a 3-class association scheme in RN−2 of size N2 /4 with the same parameters as schemes obtained from Kerdock codes (Theorem 5). D. de Caen and E. R. van Dam =-=[6]-=- constructed two infinite series of formally dual 3-class association schemes, related to Kerdock sets. We explain this formal duality by constructing two dual abelian schemes related to quaternary li... |

10 | An exponential number of generalized Kerdock codes
- Kantor
- 1982
(Show Context)
Citation Context ...ained. The claim is essentially obtained in [21], it can be proved also by method of [22]. So, it would be interesting what will happen in particular for N = 64 for X, Z, and Y . The result of Kantor =-=[19]-=- implies that if N = 2 m+1 with odd m, and if m is not a prime, then there are non-isomorphic line systems, and so there are non-isomorphic association schemes with the same parameters, i.e., the uniq... |

8 | On antipodal spherical t-designs of degree s with t ≥ 2s
- Bannai, Bannai
(Show Context)
Citation Context ...rical design and of degree s = 4. It is interesting to note that they are Q-polynomial association schemes (and not P-polynomial association scheme for N ≥ 4). The reader is referred to Bannai-Bannai =-=[3]-=- for more details, where this fact was first noticed. These association schemes are possible candidates ⎛ of universally optimal codes in the sense of Cohn-Kumar ⎞ [9]. 0 1 0 0 0 B1 = ⎜ ⎝ ⎛ B2 = ⎜ ⎝ ⎛... |

6 | Codes and association schemes: basic properties of association schemes relevant to coding - Camion - 1998 |

4 | The D4 root system is not universally optimal,
- Cohn, Conway, et al.
- 2007
(Show Context)
Citation Context ...pectively), where N must be an even power of 2. Currently all of them are possible candidates of universally optimal codes in the sense of Cohn-Kumar, at least for N ≥ 16. It is shown in Cohn et. al. =-=[10]-=- that X in R 4 (i.e., for N = 4) is not universally optimal. (It is an open question whether it is optimal or not.) Z for R 3 (i.e., for N = 4) is not universally optimal nor optimal. On the other han... |

3 |
Uniqueness of certain association schemes
- Bannai, Bannai, et al.
(Show Context)
Citation Context ...erimental results and conjectured that a some three class association scheme on 64 points determines universally optimal configuration in R 14 . This scheme is uniquely determined by their parameters =-=[2]-=- and has automorphism group 4 3 : (2 × L3(2)), where 2 × L3(2) is the stabilizer of a point. It has the following first and second eigenmatrices: P = Q = ⎛ ⎜ ⎝ 1 14 42 7 1 −6 6 −1 1 2 −2 −1 1 −2 −6 7 ... |

3 |
Kostrikin and Pham Huu Tiep, Orthogonal decompositions and integral lattices, Walter de Gruyter
- I
- 1994
(Show Context)
Citation Context ...te collection of mutually unbiased bases in C n is equivalent to an orthogonal decomposition of Lie algebra sln(C), closed under the adjoint operation. So there is a link to the well-developed theory =-=[20]-=- of orthogonal decompositions of Lie algebras. 5 Extremal line-sets and Barnes-Wall lattices In this section we discuss connections between mutually unbiased bases, extremal line-sets in R N with pres... |

2 |
Huu Tiep and P. Wocjan, Mutually unbiased bases and orthogonal decompositions of Lie algebras
- Boykin, Sitharam, et al.
- 2007
(Show Context)
Citation Context ... a prime, then there are non-isomorphic line systems, and so there are non-isomorphic association schemes with the same parameters, i.e., the uniqueness is break down. (3) Quite recently it was shown =-=[5]-=- that the problem of constructing of s pairwise mutually unbiased bases in K n (K = R or K = C) is equivalent to the problem of constructing of s Cartan subalgebras of sln(K) that are pairwise orthogo... |

2 | Sphere packings, energy minimization, and linear programming bounds - Cohn - 2005 |

1 |
Griess Jr., Pieces of 2 d : existence and uniqueness for Barnes-Wall lattices and Ypsilanti lattices
- L
- 2005
(Show Context)
Citation Context ...f size N2 + 2N in RN , as well as a 3-class association scheme of size N2 /2 in RN−1 (which corresponds to the association scheme obtained from a shortened Kerdock-like code). R. L. Griess Jr. showed =-=[15]-=- that the 64 point code and other tricosine codes can also be constructed using minimal vectors of Barnes-Wall lattice. We give later explanation of this phenomenon in terms of extremal line-sets and ... |

1 | Griess Jr., Few-cosine spherical codes and Barnes-Wall lattices, arXiv: math/0605175v1 [math.CO - L |

1 |
On extremal line set in Euclidean space with prescribed angles, Masters degree thesis
- Nakamura
- 1996
(Show Context)
Citation Context ...quely determined by the parameters. The uniqueness of Y for R 14 (i.e., for N = 16) was obtained in [2]. On the other hand, the uniqueness of X for R 16 (i.e., for N = 16) was proved by Akio Nakamura =-=[22]-=- in his masters degree thesis of Kyushu University in 1997 (it follows also from the uniqueness of the Nordstrom-Robinson code). We note that the uniqueness of Z for R 15 (i.e., for N = 16) is also ob... |