Best Known (90−27, 90, s)-Nets in Base 5
(90−27, 90, 268)-Net over F5 — Constructive and digital
Digital (63, 90, 268)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (3, 16, 16)-net over F5, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 3 and N(F) ≥ 16, using
- net from sequence [i] based on digital (3, 15)-sequence over F5, using
- digital (47, 74, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 37, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 37, 126)-net over F25, using
- digital (3, 16, 16)-net over F5, using
(90−27, 90, 711)-Net over F5 — Digital
Digital (63, 90, 711)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(590, 711, F5, 27) (dual of [711, 621, 28]-code), using
- 77 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 31 times 0) [i] based on linear OA(583, 627, F5, 27) (dual of [627, 544, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- linear OA(583, 625, F5, 27) (dual of [625, 542, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(581, 625, F5, 26) (dual of [625, 544, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(50, 2, F5, 0) (dual of [2, 2, 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(26) ⊂ Ce(25) [i] based on
- 77 step Varšamov–Edel lengthening with (ri) = (2, 1, 0, 0, 1, 5 times 0, 1, 12 times 0, 1, 21 times 0, 1, 31 times 0) [i] based on linear OA(583, 627, F5, 27) (dual of [627, 544, 28]-code), using
(90−27, 90, 86408)-Net in Base 5 — Upper bound on s
There is no (63, 90, 86409)-net in base 5, because
- 1 times m-reduction [i] would yield (63, 89, 86409)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 161 573682 754575 009133 034886 942789 701763 851612 901745 845027 176885 > 589 [i]