Best Known (71−12, 71, s)-Nets in Base 5
(71−12, 71, 13032)-Net over F5 — Constructive and digital
Digital (59, 71, 13032)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (52, 64, 13022)-net over F5, using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(557, 78125, F5, 11) (dual of [78125, 78068, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 7, F5, 0) (dual of [7, 7, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(564, 78132, F5, 12) (dual of [78132, 78068, 13]-code), using
- net defined by OOA [i] based on linear OOA(564, 13022, F5, 12, 12) (dual of [(13022, 12), 156200, 13]-NRT-code), using
- digital (1, 7, 10)-net over F5, using
(71−12, 71, 78160)-Net over F5 — Digital
Digital (59, 71, 78160)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 78160, F5, 12) (dual of [78160, 78089, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(564, 78125, F5, 12) (dual of [78125, 78061, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(536, 78125, F5, 7) (dual of [78125, 78089, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(57, 35, F5, 4) (dual of [35, 28, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(57, 44, F5, 4) (dual of [44, 37, 5]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
(71−12, 71, large)-Net in Base 5 — Upper bound on s
There is no (59, 71, large)-net in base 5, because
- 10 times m-reduction [i] would yield (59, 61, large)-net in base 5, but