Best Known (176−27, 176, s)-Nets in Base 3
(176−27, 176, 1520)-Net over F3 — Constructive and digital
Digital (149, 176, 1520)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 14, 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 (135, 162, 1513)-net over F3, using
- net defined by OOA [i] based on linear OOA(3162, 1513, F3, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- 1 times truncation [i] based on linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 19682 = 39−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- 1 times truncation [i] based on linear OA(3163, 19683, F3, 28) (dual of [19683, 19520, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(3162, 19682, F3, 27) (dual of [19682, 19520, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(3162, 19670, F3, 27) (dual of [19670, 19508, 28]-code), using
- net defined by OOA [i] based on linear OOA(3162, 1513, F3, 27, 27) (dual of [(1513, 27), 40689, 28]-NRT-code), using
- digital (1, 14, 7)-net over F3, using
(176−27, 176, 11105)-Net over F3 — Digital
Digital (149, 176, 11105)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(3176, 11105, F3, 27) (dual of [11105, 10929, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(3176, 19698, F3, 27) (dual of [19698, 19522, 28]-code), using
- construction X applied to C([0,13]) ⊂ C([1,13]) [i] based on
- linear OA(3163, 19684, F3, 27) (dual of [19684, 19521, 28]-code), using the expurgated narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [0,13], and minimum distance d ≥ |{−13,−12,…,13}|+1 = 28 (BCH-bound) [i]
- linear OA(3162, 19684, F3, 13) (dual of [19684, 19522, 14]-code), using the narrow-sense BCH-code C(I) with length 19684 | 318−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(313, 14, F3, 13) (dual of [14, 1, 14]-code or 14-arc in PG(12,3)), using
- dual of repetition code with length 14 [i]
- construction X applied to C([0,13]) ⊂ C([1,13]) [i] based on
- discarding factors / shortening the dual code based on linear OA(3176, 19698, F3, 27) (dual of [19698, 19522, 28]-code), using
(176−27, 176, 7501732)-Net in Base 3 — Upper bound on s
There is no (149, 176, 7501733)-net in base 3, because
- 1 times m-reduction [i] would yield (149, 175, 7501733)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 313487 241891 742008 976579 947888 518183 039597 418580 172195 470297 794540 598520 591657 605691 > 3175 [i]