Best Known (137−23, 137, s)-Nets in Base 3
(137−23, 137, 1790)-Net over F3 — Constructive and digital
Digital (114, 137, 1790)-net over F3, using
- 31 times duplication [i] based on digital (113, 136, 1790)-net over F3, using
- net defined by OOA [i] based on linear OOA(3136, 1790, F3, 23, 23) (dual of [(1790, 23), 41034, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3136, 19691, F3, 23) (dual of [19691, 19555, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3136, 19692, F3, 23) (dual of [19692, 19556, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3136, 19692, F3, 23) (dual of [19692, 19556, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3136, 19691, F3, 23) (dual of [19691, 19555, 24]-code), using
- net defined by OOA [i] based on linear OOA(3136, 1790, F3, 23, 23) (dual of [(1790, 23), 41034, 24]-NRT-code), using
(137−23, 137, 6881)-Net over F3 — Digital
Digital (114, 137, 6881)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3137, 6881, F3, 2, 23) (dual of [(6881, 2), 13625, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 9846, F3, 2, 23) (dual of [(9846, 2), 19555, 24]-NRT-code), using
- 31 times duplication [i] based on linear OOA(3136, 9846, F3, 2, 23) (dual of [(9846, 2), 19556, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3136, 19692, F3, 23) (dual of [19692, 19556, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3136, 19683, F3, 23) (dual of [19683, 19547, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3127, 19683, F3, 22) (dual of [19683, 19556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(30, 9, F3, 0) (dual of [9, 9, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(30, s, F3, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- OOA 2-folding [i] based on linear OA(3136, 19692, F3, 23) (dual of [19692, 19556, 24]-code), using
- 31 times duplication [i] based on linear OOA(3136, 9846, F3, 2, 23) (dual of [(9846, 2), 19556, 24]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3137, 9846, F3, 2, 23) (dual of [(9846, 2), 19555, 24]-NRT-code), using
(137−23, 137, 1945071)-Net in Base 3 — Upper bound on s
There is no (114, 137, 1945072)-net in base 3, because
- 1 times m-reduction [i] would yield (114, 136, 1945072)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 77355 736563 772252 289369 984394 335298 092725 825466 698260 333813 401025 > 3136 [i]