Best Known (59, 71, s)-Nets in Base 4
(59, 71, 2742)-Net over F4 — Constructive and digital
Digital (59, 71, 2742)-net over F4, using
- (u, u+v)-construction [i] based on
- digital (2, 8, 12)-net over F4, using
- digital (51, 63, 2730)-net over F4, using
- net defined by OOA [i] based on linear OOA(463, 2730, F4, 12, 12) (dual of [(2730, 12), 32697, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(463, 16380, F4, 12) (dual of [16380, 16317, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 16383, F4, 12) (dual of [16383, 16320, 13]-code), using
- 1 times truncation [i] based on linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using
- an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- 1 times truncation [i] based on linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(463, 16383, F4, 12) (dual of [16383, 16320, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(463, 16380, F4, 12) (dual of [16380, 16317, 13]-code), using
- net defined by OOA [i] based on linear OOA(463, 2730, F4, 12, 12) (dual of [(2730, 12), 32697, 13]-NRT-code), using
(59, 71, 16419)-Net over F4 — Digital
Digital (59, 71, 16419)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(471, 16419, F4, 12) (dual of [16419, 16348, 13]-code), using
- construction X applied to Ce(12) ⊂ Ce(6) [i] based on
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(436, 16384, F4, 7) (dual of [16384, 16348, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(47, 35, F4, 4) (dual of [35, 28, 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
(59, 71, large)-Net in Base 4 — Upper bound on s
There is no (59, 71, large)-net in base 4, because
- 10 times m-reduction [i] would yield (59, 61, large)-net in base 4, but