Best Known (81−16, 81, s)-Nets in Base 5
(81−16, 81, 1959)-Net over F5 — Constructive and digital
Digital (65, 81, 1959)-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 (57, 73, 1953)-net over F5, using
- net defined by OOA [i] based on linear OOA(573, 1953, F5, 16, 16) (dual of [(1953, 16), 31175, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(573, 15624, F5, 16) (dual of [15624, 15551, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [i]
- discarding factors / shortening the dual code based on linear OA(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(573, 15624, F5, 16) (dual of [15624, 15551, 17]-code), using
- net defined by OOA [i] based on linear OOA(573, 1953, F5, 16, 16) (dual of [(1953, 16), 31175, 17]-NRT-code), using
- digital (0, 8, 6)-net over F5, using
(81−16, 81, 14902)-Net over F5 — Digital
Digital (65, 81, 14902)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(581, 14902, F5, 16) (dual of [14902, 14821, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(581, 15634, F5, 16) (dual of [15634, 15553, 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(573, 15625, F5, 16) (dual of [15625, 15552, 17]-code), using
- an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, 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(581, 15634, F5, 16) (dual of [15634, 15553, 17]-code), using
(81−16, 81, large)-Net in Base 5 — Upper bound on s
There is no (65, 81, large)-net in base 5, because
- 14 times m-reduction [i] would yield (65, 67, large)-net in base 5, but