Best Known (77, 77+12, s)-Nets in Base 4
(77, 77+12, 43702)-Net over F4 — Constructive and digital
Digital (77, 89, 43702)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (69, 81, 43690)-net over F4, using
- net defined by OOA [i] based on linear OOA(481, 43690, F4, 12, 12) (dual of [(43690, 12), 524199, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(481, 262140, F4, 12) (dual of [262140, 262059, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 262143, F4, 12) (dual of [262143, 262062, 13]-code), using
- 1 times truncation [i] based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 1 times truncation [i] based on linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(481, 262143, F4, 12) (dual of [262143, 262062, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(481, 262140, F4, 12) (dual of [262140, 262059, 13]-code), using
- net defined by OOA [i] based on linear OOA(481, 43690, F4, 12, 12) (dual of [(43690, 12), 524199, 13]-NRT-code), using
(77, 77+12, 262187)-Net over F4 — Digital
Digital (77, 89, 262187)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 262187, F4, 12) (dual of [262187, 262098, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- linear OA(482, 262144, F4, 13) (dual of [262144, 262062, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(446, 262144, F4, 7) (dual of [262144, 262098, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 49−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(47, 43, F4, 4) (dual of [43, 36, 5]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
(77, 77+12, large)-Net in Base 4 — Upper bound on s
There is no (77, 89, large)-net in base 4, because
- 10 times m-reduction [i] would yield (77, 79, large)-net in base 4, but