Best Known (12, 12+8, s)-Nets in Base 25
(12, 12+8, 650)-Net over F25 — Constructive and digital
Digital (12, 20, 650)-net over F25, using
- generalized (u, u+v)-construction [i] based on
- digital (0, 0, 26)-net over F25, using
- s-reduction based on digital (0, 0, s)-net over F25 with arbitrarily large s, using
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- digital (0, 0, 26)-net over F25 (see above)
- 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, 1, 26)-net over F25 (see above)
- digital (0, 1, 26)-net over F25 (see above)
- digital (0, 1, 26)-net over F25 (see above)
- digital (0, 2, 26)-net over F25, using
- digital (0, 2, 26)-net over F25 (see above)
- 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 (0, 8, 26)-net over F25, using
- net from sequence [i] based on digital (0, 25)-sequence over F25 (see above)
- digital (0, 0, 26)-net over F25, using
(12, 12+8, 1400)-Net over F25 — Digital
Digital (12, 20, 1400)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2520, 1400, F25, 8) (dual of [1400, 1380, 9]-code), using
- 767 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0, 1, 62 times 0, 1, 215 times 0, 1, 476 times 0) [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
- construction XX applied to C1 = C([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,5}, and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2513, 624, F25, 7) (dual of [624, 611, 8]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,6], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(2515, 624, F25, 8) (dual of [624, 609, 9]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,6}, and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2511, 624, F25, 6) (dual of [624, 613, 7]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,5], and designed minimum distance d ≥ |I|+1 = 7 [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([623,5]), C2 = C([0,6]), C3 = C1 + C2 = C([0,5]), and C∩ = C1 ∩ C2 = C([623,6]) [i] based on
- 767 step Varšamov–Edel lengthening with (ri) = (2, 10 times 0, 1, 62 times 0, 1, 215 times 0, 1, 476 times 0) [i] based on linear OA(2515, 628, F25, 8) (dual of [628, 613, 9]-code), using
(12, 12+8, 900618)-Net in Base 25 — Upper bound on s
There is no (12, 20, 900619)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 9094 971853 142145 555476 061985 > 2520 [i]