Best Known (222−31, 222, s)-Nets in Base 3
(222−31, 222, 11810)-Net over F3 — Constructive and digital
Digital (191, 222, 11810)-net over F3, using
- net defined by OOA [i] based on linear OOA(3222, 11810, F3, 31, 31) (dual of [(11810, 31), 365888, 32]-NRT-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3222, 177151, F3, 31) (dual of [177151, 176929, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 177159, F3, 31) (dual of [177159, 176937, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 177159, F3, 31) (dual of [177159, 176937, 32]-code), using
- OOA 15-folding and stacking with additional row [i] based on linear OA(3222, 177151, F3, 31) (dual of [177151, 176929, 32]-code), using
(222−31, 222, 44289)-Net over F3 — Digital
Digital (191, 222, 44289)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(3222, 44289, F3, 4, 31) (dual of [(44289, 4), 176934, 32]-NRT-code), using
- OOA 4-folding [i] based on linear OA(3222, 177156, F3, 31) (dual of [177156, 176934, 32]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 177159, F3, 31) (dual of [177159, 176937, 32]-code), using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- linear OA(3221, 177147, F3, 31) (dual of [177147, 176926, 32]-code), using an extension Ce(30) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,30], and designed minimum distance d ≥ |I|+1 = 31 [i]
- linear OA(3210, 177147, F3, 29) (dual of [177147, 176937, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(31, 12, F3, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(31, s, F3, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(30) ⊂ Ce(28) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 177159, F3, 31) (dual of [177159, 176937, 32]-code), using
- OOA 4-folding [i] based on linear OA(3222, 177156, F3, 31) (dual of [177156, 176934, 32]-code), using
(222−31, 222, large)-Net in Base 3 — Upper bound on s
There is no (191, 222, large)-net in base 3, because
- 29 times m-reduction [i] would yield (191, 193, large)-net in base 3, but