Best Known (158, 158+21, s)-Nets in Base 3
(158, 158+21, 53150)-Net over F3 — Constructive and digital
Digital (158, 179, 53150)-net over F3, using
- (u, u+v)-construction [i] based on
- digital (1, 11, 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 (147, 168, 53143)-net over F3, using
- net defined by OOA [i] based on linear OOA(3168, 53143, F3, 21, 21) (dual of [(53143, 21), 1115835, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3168, 531431, F3, 21) (dual of [531431, 531263, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 531440, F3, 21) (dual of [531440, 531272, 22]-code), using
- 1 times truncation [i] based on linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using
- an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- 1 times truncation [i] based on linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(3168, 531440, F3, 21) (dual of [531440, 531272, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3168, 531431, F3, 21) (dual of [531431, 531263, 22]-code), using
- net defined by OOA [i] based on linear OOA(3168, 53143, F3, 21, 21) (dual of [(53143, 21), 1115835, 22]-NRT-code), using
- digital (1, 11, 7)-net over F3, using
(158, 158+21, 185675)-Net over F3 — Digital
Digital (158, 179, 185675)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3179, 185675, F3, 2, 21) (dual of [(185675, 2), 371171, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3179, 265749, F3, 2, 21) (dual of [(265749, 2), 531319, 22]-NRT-code), using
- 1 step truncation [i] based on linear OOA(3180, 265750, F3, 2, 22) (dual of [(265750, 2), 531320, 23]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3180, 531500, F3, 22) (dual of [531500, 531320, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(15) [i] based on
- linear OA(3169, 531441, F3, 22) (dual of [531441, 531272, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(3121, 531441, F3, 16) (dual of [531441, 531320, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 531440 = 312−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(311, 59, F3, 5) (dual of [59, 48, 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(21) ⊂ Ce(15) [i] based on
- OOA 2-folding [i] based on linear OA(3180, 531500, F3, 22) (dual of [531500, 531320, 23]-code), using
- 1 step truncation [i] based on linear OOA(3180, 265750, F3, 2, 22) (dual of [(265750, 2), 531320, 23]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(3179, 265749, F3, 2, 21) (dual of [(265749, 2), 531319, 22]-NRT-code), using
(158, 158+21, large)-Net in Base 3 — Upper bound on s
There is no (158, 179, large)-net in base 3, because
- 19 times m-reduction [i] would yield (158, 160, large)-net in base 3, but