Best Known (113, 113+21, s)-Nets in Base 3
(113, 113+21, 1971)-Net over F3 — Constructive and digital
Digital (113, 134, 1971)-net over F3, using
- 31 times duplication [i] based on digital (112, 133, 1971)-net over F3, using
- net defined by OOA [i] based on linear OOA(3133, 1971, F3, 21, 21) (dual of [(1971, 21), 41258, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3133, 19711, F3, 21) (dual of [19711, 19578, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(3133, 19716, F3, 21) (dual of [19716, 19583, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 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(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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(3133, 19716, F3, 21) (dual of [19716, 19583, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(3133, 19711, F3, 21) (dual of [19711, 19578, 22]-code), using
- net defined by OOA [i] based on linear OOA(3133, 1971, F3, 21, 21) (dual of [(1971, 21), 41258, 22]-NRT-code), using
(113, 113+21, 9858)-Net over F3 — Digital
Digital (113, 134, 9858)-net over F3, using
- 31 times duplication [i] based on digital (112, 133, 9858)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 9858, F3, 2, 21) (dual of [(9858, 2), 19583, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3133, 19716, F3, 21) (dual of [19716, 19583, 22]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 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(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(36, 33, F3, 3) (dual of [33, 27, 4]-code or 33-cap in PG(5,3)), using
- discarding factors / shortening the dual code based on linear OA(36, 48, F3, 3) (dual of [48, 42, 4]-code or 48-cap in PG(5,3)), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- OOA 2-folding [i] based on linear OA(3133, 19716, F3, 21) (dual of [19716, 19583, 22]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3133, 9858, F3, 2, 21) (dual of [(9858, 2), 19583, 22]-NRT-code), using
(113, 113+21, 5019473)-Net in Base 3 — Upper bound on s
There is no (113, 134, 5019474)-net in base 3, because
- 1 times m-reduction [i] would yield (113, 133, 5019474)-net in base 3, but
- the generalized Rao bound for nets shows that 3m ≥ 2865 017198 648052 575897 338919 662313 894031 710177 519534 948736 559301 > 3133 [i]