Best Known (68, 80, s)-Nets in Base 4
(68, 80, 10934)-Net over F4 — Constructive and digital
Digital (68, 80, 10934)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (60, 72, 10922)-net over F4, using
- net defined by OOA [i] based on linear OOA(472, 10922, F4, 12, 12) (dual of [(10922, 12), 130992, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(472, 65532, F4, 12) (dual of [65532, 65460, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(472, 65535, F4, 12) (dual of [65535, 65463, 13]-code), using
- 1 times truncation [i] based on 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]
- 1 times truncation [i] based on linear OA(473, 65536, F4, 13) (dual of [65536, 65463, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(472, 65535, F4, 12) (dual of [65535, 65463, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(472, 65532, F4, 12) (dual of [65532, 65460, 13]-code), using
- net defined by OOA [i] based on linear OOA(472, 10922, F4, 12, 12) (dual of [(10922, 12), 130992, 13]-NRT-code), using
(68, 80, 65575)-Net over F4 — Digital
Digital (68, 80, 65575)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(480, 65575, F4, 12) (dual of [65575, 65495, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- 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(441, 65536, F4, 7) (dual of [65536, 65495, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 65535 = 48−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(47, 39, F4, 4) (dual of [39, 32, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
(68, 80, large)-Net in Base 4 — Upper bound on s
There is no (68, 80, large)-net in base 4, because
- 10 times m-reduction [i] would yield (68, 70, large)-net in base 4, but