Best Known (33−14, 33, s)-Nets in Base 25
(33−14, 33, 158)-Net over F25 — Constructive and digital
Digital (19, 33, 158)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 4, 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 (4, 11, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (4, 18, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25 (see above)
- digital (0, 4, 26)-net over F25, using
(33−14, 33, 859)-Net over F25 — Digital
Digital (19, 33, 859)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2533, 859, F25, 14) (dual of [859, 826, 15]-code), using
- 225 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 20 times 0, 1, 61 times 0, 1, 136 times 0) [i] based on linear OA(2527, 628, F25, 14) (dual of [628, 601, 15]-code), using
- construction XX applied to C1 = C([623,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([623,12]) [i] based on
- 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(2525, 624, F25, 13) (dual of [624, 599, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2527, 624, F25, 14) (dual of [624, 597, 15]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [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(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([623,11]), C2 = C([0,12]), C3 = C1 + C2 = C([0,11]), and C∩ = C1 ∩ C2 = C([623,12]) [i] based on
- 225 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 20 times 0, 1, 61 times 0, 1, 136 times 0) [i] based on linear OA(2527, 628, F25, 14) (dual of [628, 601, 15]-code), using
(33−14, 33, 548268)-Net in Base 25 — Upper bound on s
There is no (19, 33, 548269)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 13552 588231 343721 886280 517678 337607 290618 077001 > 2533 [i]