Best Known (91, 91+18, s)-Nets in Base 5
(91, 91+18, 8691)-Net over F5 — Constructive and digital
Digital (91, 109, 8691)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 10, 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
- a shift-net [i]
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (81, 99, 8681)-net over F5, using
- net defined by OOA [i] based on linear OOA(599, 8681, F5, 18, 18) (dual of [(8681, 18), 156159, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(599, 78129, F5, 18) (dual of [78129, 78030, 19]-code), using
- discarding factors / shortening the dual code based on linear OA(599, 78132, F5, 18) (dual of [78132, 78033, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(16) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(592, 78125, F5, 17) (dual of [78125, 78033, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [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(17) ⊂ Ce(16) [i] based on
- discarding factors / shortening the dual code based on linear OA(599, 78132, F5, 18) (dual of [78132, 78033, 19]-code), using
- OA 9-folding and stacking [i] based on linear OA(599, 78129, F5, 18) (dual of [78129, 78030, 19]-code), using
- net defined by OOA [i] based on linear OOA(599, 8681, F5, 18, 18) (dual of [(8681, 18), 156159, 19]-NRT-code), using
- digital (1, 10, 10)-net over F5, using
(91, 91+18, 78171)-Net over F5 — Digital
Digital (91, 109, 78171)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 78171, F5, 18) (dual of [78171, 78062, 19]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(5108, 78169, F5, 18) (dual of [78169, 78061, 19]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(599, 78125, F5, 18) (dual of [78125, 78026, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- 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(59, 44, F5, 5) (dual of [44, 35, 6]-code), using
- construction X applied to Ce(17) ⊂ Ce(11) [i] based on
- linear OA(5108, 78170, F5, 17) (dual of [78170, 78062, 18]-code), using Gilbert–Varšamov bound and bm = 5108 > Vbs−1(k−1) = 398 335166 406147 712073 538102 780263 876458 188250 106232 251485 107505 120840 993573 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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
- linear OA(5108, 78169, F5, 18) (dual of [78169, 78061, 19]-code), using
- construction X with Varšamov bound [i] based on
(91, 91+18, large)-Net in Base 5 — Upper bound on s
There is no (91, 109, large)-net in base 5, because
- 16 times m-reduction [i] would yield (91, 93, large)-net in base 5, but