Best Known (68, 68+14, s)-Nets in Base 4
(68, 68+14, 9363)-Net over F4 — Constructive and digital
Digital (68, 82, 9363)-net over F4, using
- 41 times duplication [i] based on digital (67, 81, 9363)-net over F4, using
- net defined by OOA [i] based on linear OOA(481, 9363, F4, 14, 14) (dual of [(9363, 14), 131001, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(481, 65541, F4, 14) (dual of [65541, 65460, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- discarding factors / shortening the dual code based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- OA 7-folding and stacking [i] based on linear OA(481, 65541, F4, 14) (dual of [65541, 65460, 15]-code), using
- net defined by OOA [i] based on linear OOA(481, 9363, F4, 14, 14) (dual of [(9363, 14), 131001, 15]-NRT-code), using
(68, 68+14, 32772)-Net over F4 — Digital
Digital (68, 82, 32772)-net over F4, using
- 41 times duplication [i] based on digital (67, 81, 32772)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(481, 32772, F4, 2, 14) (dual of [(32772, 2), 65463, 15]-NRT-code), using
- OOA 2-folding [i] based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- linear OA(481, 65536, F4, 14) (dual of [65536, 65455, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(40, 8, F4, 0) (dual of [8, 8, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(40, s, F4, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(13) ⊂ Ce(12) [i] based on
- OOA 2-folding [i] based on linear OA(481, 65544, F4, 14) (dual of [65544, 65463, 15]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(481, 32772, F4, 2, 14) (dual of [(32772, 2), 65463, 15]-NRT-code), using
(68, 68+14, large)-Net in Base 4 — Upper bound on s
There is no (68, 82, large)-net in base 4, because
- 12 times m-reduction [i] would yield (68, 70, large)-net in base 4, but