Best Known (45, 45+13, s)-Nets in Base 8
(45, 45+13, 5463)-Net over F8 — Constructive and digital
Digital (45, 58, 5463)-net over F8, using
- 81 times duplication [i] based on digital (44, 57, 5463)-net over F8, using
- net defined by OOA [i] based on linear OOA(857, 5463, F8, 13, 13) (dual of [(5463, 13), 70962, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(857, 32779, F8, 13) (dual of [32779, 32722, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(856, 32768, F8, 13) (dual of [32768, 32712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(81, 11, F8, 1) (dual of [11, 10, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(857, 32779, F8, 13) (dual of [32779, 32722, 14]-code), using
- net defined by OOA [i] based on linear OOA(857, 5463, F8, 13, 13) (dual of [(5463, 13), 70962, 14]-NRT-code), using
(45, 45+13, 32781)-Net over F8 — Digital
Digital (45, 58, 32781)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(858, 32781, F8, 13) (dual of [32781, 32723, 14]-code), using
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
- linear OA(856, 32768, F8, 13) (dual of [32768, 32712, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(846, 32768, F8, 11) (dual of [32768, 32722, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(841, 32768, F8, 10) (dual of [32768, 32727, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(12) ⊂ Ce(10) ⊂ Ce(9) [i] based on
(45, 45+13, large)-Net in Base 8 — Upper bound on s
There is no (45, 58, large)-net in base 8, because
- 11 times m-reduction [i] would yield (45, 47, large)-net in base 8, but