Best Known (148−23, 148, s)-Nets in Base 3
(148−23, 148, 1797)-Net over F3 — Constructive and digital
Digital (125, 148, 1797)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 7)-net over F3, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F3 with g(F) = 1 and N(F) ≥ 7, using
- net from sequence [i] based on digital (1, 6)-sequence over F3, using
- 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
- digital (1, 12, 7)-net over F3, using
(148−23, 148, 9865)-Net over F3 — Digital
Digital (125, 148, 9865)-net over F3, using
- 31 times duplication [i] based on digital (124, 147, 9865)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 9865, F3, 2, 23) (dual of [(9865, 2), 19583, 24]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3147, 19730, F3, 23) (dual of [19730, 19583, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(16) [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(3100, 19683, F3, 17) (dual of [19683, 19583, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(311, 47, F3, 5) (dual of [47, 36, 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 Ce(22) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3147, 19730, F3, 23) (dual of [19730, 19583, 24]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3147, 9865, F3, 2, 23) (dual of [(9865, 2), 19583, 24]-NRT-code), using
(148−23, 148, 5835234)-Net in Base 3 — Upper bound on s
There is no (125, 148, 5835235)-net in base 3, because
- 1 times m-reduction [i] would yield (125, 147, 5835235)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 13703 285002 586862 218993 911244 003486 293222 190498 305756 463071 874028 933531 > 3147 [i]