Best Known (234−31, 234, s)-Nets in Base 3
(234−31, 234, 11813)-Net over F3 — Constructive and digital
Digital (203, 234, 11813)-net over F3, using
- 32 times duplication [i] based on digital (201, 232, 11813)-net over F3, using
- net defined by OOA [i] based on linear OOA(3232, 11813, F3, 31, 31) (dual of [(11813, 31), 365971, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3232, 177196, F3, 31) (dual of [177196, 176964, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3232, 177203, F3, 31) (dual of [177203, 176971, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(311, 55, F3, 5) (dual of [55, 44, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(311, 85, F3, 5) (dual of [85, 74, 6]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3232, 177203, F3, 31) (dual of [177203, 176971, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3232, 177196, F3, 31) (dual of [177196, 176964, 32]-code), using
- net defined by OOA [i] based on linear OOA(3232, 11813, F3, 31, 31) (dual of [(11813, 31), 365971, 32]-NRT-code), using
(234−31, 234, 59068)-Net over F3 — Digital
Digital (203, 234, 59068)-net over F3, using
- 31 times duplication [i] based on digital (202, 233, 59068)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 59068, F3, 3, 31) (dual of [(59068, 3), 176971, 32]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3233, 177204, F3, 31) (dual of [177204, 176971, 32]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- linear OA(3221, 177148, F3, 31) (dual of [177148, 176927, 32]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,15], and minimum distance d ≥ |{−15,−14,…,15}|+1 = 32 (BCH-bound) [i]
- linear OA(3177, 177148, F3, 25) (dual of [177148, 176971, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 177148 | 322−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(312, 56, F3, 5) (dual of [56, 44, 6]-code), using
- (u, u+v)-construction [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- Hamming code H(4,3) [i]
- discarding factors / shortening the dual code based on linear OA(34, 40, F3, 2) (dual of [40, 36, 3]-code), using
- linear OA(38, 28, F3, 5) (dual of [28, 20, 6]-code), using
- dual code (with bound on d by construction Y1) [i] based on
- linear OA(34, 28, F3, 2) (dual of [28, 24, 3]-code), using
- (u, u+v)-construction [i] based on
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- OOA 3-folding [i] based on linear OA(3233, 177204, F3, 31) (dual of [177204, 176971, 32]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3233, 59068, F3, 3, 31) (dual of [(59068, 3), 176971, 32]-NRT-code), using
(234−31, 234, large)-Net in Base 3 — Upper bound on s
There is no (203, 234, large)-net in base 3, because
- 29 times m-reduction [i] would yield (203, 205, large)-net in base 3, but