Best Known (55−29, 55, s)-Nets in Base 25
(55−29, 55, 200)-Net over F25 — Constructive and digital
Digital (26, 55, 200)-net over F25, using
- t-expansion [i] based on digital (25, 55, 200)-net over F25, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 25 and N(F) ≥ 200, using
- net from sequence [i] based on digital (25, 199)-sequence over F25, using
(55−29, 55, 298)-Net over F25 — Digital
Digital (26, 55, 298)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2555, 298, F25, 2, 29) (dual of [(298, 2), 541, 30]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2555, 315, F25, 2, 29) (dual of [(315, 2), 575, 30]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2555, 630, F25, 29) (dual of [630, 575, 30]-code), using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- linear OA(2554, 625, F25, 29) (dual of [625, 571, 30]-code), using an extension Ce(28) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,28], and designed minimum distance d ≥ |I|+1 = 29 [i]
- linear OA(2550, 625, F25, 27) (dual of [625, 575, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(251, 5, F25, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(251, s, F25, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(28) ⊂ Ce(26) [i] based on
- OOA 2-folding [i] based on linear OA(2555, 630, F25, 29) (dual of [630, 575, 30]-code), using
- discarding factors / shortening the dual code based on linear OOA(2555, 315, F25, 2, 29) (dual of [(315, 2), 575, 30]-NRT-code), using
(55−29, 55, 62122)-Net in Base 25 — Upper bound on s
There is no (26, 55, 62123)-net in base 25, because
- 1 times m-reduction [i] would yield (26, 54, 62123)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 3081 598957 782232 704214 216062 410164 235387 399307 707309 492851 351856 057848 245745 > 2554 [i]