Best Known (31−13, 31, s)-Nets in Base 25
(31−13, 31, 182)-Net over F25 — Constructive and digital
Digital (18, 31, 182)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 1, 26)-net over F25, using
- s-reduction based on digital (0, 1, s)-net over F25 with arbitrarily large s, using
- digital (0, 2, 26)-net over F25, using
- digital (0, 2, 26)-net over F25 (see above)
- digital (0, 3, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- generalized Faure sequence [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 0 and N(F) ≥ 26, using
- the rational function field F25(x) [i]
- Niederreiter sequence [i]
- net from sequence [i] based on digital (0, 25)-sequence over F25, using
- digital (0, 4, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25 (see above)
- digital (0, 6, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25 (see above)
- digital (0, 13, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25 (see above)
- digital (0, 1, 26)-net over F25, using
(31−13, 31, 917)-Net over F25 — Digital
Digital (18, 31, 917)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2531, 917, F25, 13) (dual of [917, 886, 14]-code), using
- 283 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 23 times 0, 1, 80 times 0, 1, 172 times 0) [i] based on linear OA(2525, 628, F25, 13) (dual of [628, 603, 14]-code), using
- construction XX applied to C1 = C([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- linear OA(2523, 624, F25, 12) (dual of [624, 601, 13]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2523, 624, F25, 12) (dual of [624, 601, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2525, 624, F25, 13) (dual of [624, 599, 14]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2521, 624, F25, 11) (dual of [624, 603, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(250, 2, F25, 0) (dual of [2, 2, 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
- linear OA(250, 2, F25, 0) (dual of [2, 2, 1]-code) (see above)
- construction XX applied to C1 = C([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- 283 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 23 times 0, 1, 80 times 0, 1, 172 times 0) [i] based on linear OA(2525, 628, F25, 13) (dual of [628, 603, 14]-code), using
(31−13, 31, 1218175)-Net in Base 25 — Upper bound on s
There is no (18, 31, 1218176)-net in base 25, because
- 1 times m-reduction [i] would yield (18, 30, 1218176)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 867363 355446 347451 193379 701895 781861 709825 > 2530 [i]