Best Known (230−30, 230, s)-Nets in Base 3
(230−30, 230, 11813)-Net over F3 — Constructive and digital
Digital (200, 230, 11813)-net over F3, using
- net defined by OOA [i] based on linear OOA(3230, 11813, F3, 30, 30) (dual of [(11813, 30), 354160, 31]-NRT-code), using
- OA 15-folding and stacking [i] based on linear OA(3230, 177195, F3, 30) (dual of [177195, 176965, 31]-code), using
- discarding factors / shortening the dual code based on linear OA(3230, 177201, F3, 30) (dual of [177201, 176971, 31]-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(39, 53, F3, 4) (dual of [53, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-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(3230, 177201, F3, 30) (dual of [177201, 176971, 31]-code), using
- OA 15-folding and stacking [i] based on linear OA(3230, 177195, F3, 30) (dual of [177195, 176965, 31]-code), using
(230−30, 230, 59067)-Net over F3 — Digital
Digital (200, 230, 59067)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3230, 59067, F3, 3, 30) (dual of [(59067, 3), 176971, 31]-NRT-code), using
- OOA 3-folding [i] based on linear OA(3230, 177201, F3, 30) (dual of [177201, 176971, 31]-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(39, 53, F3, 4) (dual of [53, 44, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 80 = 34−1, defining interval I = [0,2], and designed minimum distance d ≥ |I|+1 = 5 [i]
- discarding factors / shortening the dual code based on linear OA(39, 80, F3, 4) (dual of [80, 71, 5]-code), using
- construction X applied to C([0,15]) ⊂ C([0,12]) [i] based on
- OOA 3-folding [i] based on linear OA(3230, 177201, F3, 30) (dual of [177201, 176971, 31]-code), using
(230−30, 230, large)-Net in Base 3 — Upper bound on s
There is no (200, 230, large)-net in base 3, because
- 28 times m-reduction [i] would yield (200, 202, large)-net in base 3, but