Best Known (231−30, 231, s)-Nets in Base 3
(231−30, 231, 11813)-Net over F3 — Constructive and digital
Digital (201, 231, 11813)-net over F3, using
- 1 times m-reduction [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
(231−30, 231, 60798)-Net over F3 — Digital
Digital (201, 231, 60798)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3231, 60798, F3, 2, 30) (dual of [(60798, 2), 121365, 31]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3231, 88601, F3, 2, 30) (dual of [(88601, 2), 176971, 31]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3231, 177202, F3, 30) (dual of [177202, 176971, 31]-code), using
- 1 times truncation [i] 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
- 1 times truncation [i] based on linear OA(3232, 177203, F3, 31) (dual of [177203, 176971, 32]-code), using
- OOA 2-folding [i] based on linear OA(3231, 177202, F3, 30) (dual of [177202, 176971, 31]-code), using
- discarding factors / shortening the dual code based on linear OOA(3231, 88601, F3, 2, 30) (dual of [(88601, 2), 176971, 31]-NRT-code), using
(231−30, 231, large)-Net in Base 3 — Upper bound on s
There is no (201, 231, large)-net in base 3, because
- 28 times m-reduction [i] would yield (201, 203, large)-net in base 3, but