Best Known (91−12, 91, s)-Nets in Base 3
(91−12, 91, 29528)-Net over F3 — Constructive and digital
Digital (79, 91, 29528)-net over F3, using
- 31 times duplication [i] based on digital (78, 90, 29528)-net over F3, using
- net defined by OOA [i] based on linear OOA(390, 29528, F3, 12, 12) (dual of [(29528, 12), 354246, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(390, 177168, F3, 12) (dual of [177168, 177078, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(390, 177170, F3, 12) (dual of [177170, 177080, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(31, 23, F3, 1) (dual of [23, 22, 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(12) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(390, 177170, F3, 12) (dual of [177170, 177080, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(390, 177168, F3, 12) (dual of [177168, 177078, 13]-code), using
- net defined by OOA [i] based on linear OOA(390, 29528, F3, 12, 12) (dual of [(29528, 12), 354246, 13]-NRT-code), using
(91−12, 91, 88586)-Net over F3 — Digital
Digital (79, 91, 88586)-net over F3, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(391, 88586, F3, 2, 12) (dual of [(88586, 2), 177081, 13]-NRT-code), using
- OOA 2-folding [i] based on linear OA(391, 177172, F3, 12) (dual of [177172, 177081, 13]-code), using
- 1 times code embedding in larger space [i] based on linear OA(390, 177171, F3, 12) (dual of [177171, 177081, 13]-code), using
- construction X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(389, 177147, F3, 13) (dual of [177147, 177058, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(367, 177147, F3, 10) (dual of [177147, 177080, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 177146 = 311−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(323, 24, F3, 23) (dual of [24, 1, 24]-code or 24-arc in PG(22,3)), using
- dual of repetition code with length 24 [i]
- linear OA(31, 24, F3, 1) (dual of [24, 23, 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 X4 applied to Ce(12) ⊂ Ce(9) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(390, 177171, F3, 12) (dual of [177171, 177081, 13]-code), using
- OOA 2-folding [i] based on linear OA(391, 177172, F3, 12) (dual of [177172, 177081, 13]-code), using
(91−12, 91, large)-Net in Base 3 — Upper bound on s
There is no (79, 91, large)-net in base 3, because
- 10 times m-reduction [i] would yield (79, 81, large)-net in base 3, but