Best Known (33, 33+27, s)-Nets in Base 25
(33, 33+27, 252)-Net over F25 — Constructive and digital
Digital (33, 60, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 23, 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
- digital (10, 37, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 23, 126)-net over F25, using
(33, 33+27, 769)-Net over F25 — Digital
Digital (33, 60, 769)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2560, 769, F25, 27) (dual of [769, 709, 28]-code), using
- 133 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 35 times 0, 1, 59 times 0) [i] based on linear OA(2550, 626, F25, 27) (dual of [626, 576, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 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(2549, 625, F25, 26) (dual of [625, 576, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(250, 1, F25, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(250, s, F25, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(26) ⊂ Ce(25) [i] based on
- 133 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0, 1, 35 times 0, 1, 59 times 0) [i] based on linear OA(2550, 626, F25, 27) (dual of [626, 576, 28]-code), using
(33, 33+27, 522019)-Net in Base 25 — Upper bound on s
There is no (33, 60, 522020)-net in base 25, because
- 1 times m-reduction [i] would yield (33, 59, 522020)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 30093 273053 587559 723935 378253 891025 859305 744585 966479 193655 341549 230969 027245 024993 > 2559 [i]