Best Known (57−13, 57, s)-Nets in Base 8
(57−13, 57, 5463)-Net over F8 — Constructive and digital
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
(57−13, 57, 27757)-Net over F8 — Digital
Digital (44, 57, 27757)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(857, 27757, F8, 13) (dual of [27757, 27700, 14]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(857, 32779, F8, 13) (dual of [32779, 32722, 14]-code), using
(57−13, 57, large)-Net in Base 8 — Upper bound on s
There is no (44, 57, large)-net in base 8, because
- 11 times m-reduction [i] would yield (44, 46, large)-net in base 8, but