Best Known (46−13, 46, s)-Nets in Base 8
(46−13, 46, 684)-Net over F8 — Constructive and digital
Digital (33, 46, 684)-net over F8, using
- net defined by OOA [i] based on linear OOA(846, 684, F8, 13, 13) (dual of [(684, 13), 8846, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(846, 4105, F8, 13) (dual of [4105, 4059, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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(846, 4105, F8, 13) (dual of [4105, 4059, 14]-code), using
(46−13, 46, 3464)-Net over F8 — Digital
Digital (33, 46, 3464)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(846, 3464, F8, 13) (dual of [3464, 3418, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(846, 4105, F8, 13) (dual of [4105, 4059, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(10) [i] based on
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(81, 9, F8, 1) (dual of [9, 8, 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(846, 4105, F8, 13) (dual of [4105, 4059, 14]-code), using
(46−13, 46, 2536870)-Net in Base 8 — Upper bound on s
There is no (33, 46, 2536871)-net in base 8, because
- 1 times m-reduction [i] would yield (33, 45, 2536871)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 43556 151056 274723 586120 964188 718894 035193 > 845 [i]