Best Known (77, 77+16, s)-Nets in Base 5
(77, 77+16, 9771)-Net over F5 — Constructive and digital
Digital (77, 93, 9771)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (0, 8, 6)-net over F5, using
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 0 and N(F) ≥ 6, using
- the rational function field F5(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 5)-sequence over F5, using
- digital (69, 85, 9765)-net over F5, using
- net defined by OOA [i] based on linear OOA(585, 9765, F5, 16, 16) (dual of [(9765, 16), 156155, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(585, 78120, F5, 16) (dual of [78120, 78035, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(585, 78120, F5, 16) (dual of [78120, 78035, 17]-code), using
- net defined by OOA [i] based on linear OOA(585, 9765, F5, 16, 16) (dual of [(9765, 16), 156155, 17]-NRT-code), using
- digital (0, 8, 6)-net over F5, using
(77, 77+16, 59232)-Net over F5 — Digital
Digital (77, 93, 59232)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(593, 59232, F5, 16) (dual of [59232, 59139, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(593, 78134, F5, 16) (dual of [78134, 78041, 17]-code), using
- (u, u+v)-construction [i] based on
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- dual of repetition code with length 9 [i]
- linear OA(585, 78125, F5, 16) (dual of [78125, 78040, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 78124 = 57−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(58, 9, F5, 8) (dual of [9, 1, 9]-code or 9-arc in PG(7,5)), using
- (u, u+v)-construction [i] based on
- discarding factors / shortening the dual code based on linear OA(593, 78134, F5, 16) (dual of [78134, 78041, 17]-code), using
(77, 77+16, large)-Net in Base 5 — Upper bound on s
There is no (77, 93, large)-net in base 5, because
- 14 times m-reduction [i] would yield (77, 79, large)-net in base 5, but