Best Known (68−32, 68, s)-Nets in Base 25
(68−32, 68, 252)-Net over F25 — Constructive and digital
Digital (36, 68, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 26, 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, 42, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 26, 126)-net over F25, using
(68−32, 68, 651)-Net over F25 — Digital
Digital (36, 68, 651)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2568, 651, F25, 32) (dual of [651, 583, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 673, F25, 32) (dual of [673, 605, 33]-code), using
- 37 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0) [i] based on linear OA(2560, 628, F25, 32) (dual of [628, 568, 33]-code), using
- construction XX applied to C1 = C([623,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([623,30]) [i] based on
- linear OA(2558, 624, F25, 31) (dual of [624, 566, 32]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,29}, and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2558, 624, F25, 31) (dual of [624, 566, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [i]
- linear OA(2560, 624, F25, 32) (dual of [624, 564, 33]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2556, 624, F25, 30) (dual of [624, 568, 31]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,29], and designed minimum distance d ≥ |I|+1 = 31 [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([623,29]), C2 = C([0,30]), C3 = C1 + C2 = C([0,29]), and C∩ = C1 ∩ C2 = C([623,30]) [i] based on
- 37 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 19 times 0) [i] based on linear OA(2560, 628, F25, 32) (dual of [628, 568, 33]-code), using
- discarding factors / shortening the dual code based on linear OA(2568, 673, F25, 32) (dual of [673, 605, 33]-code), using
(68−32, 68, 247490)-Net in Base 25 — Upper bound on s
There is no (36, 68, 247491)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 114799 534685 392815 953705 071044 931894 484973 468748 591315 487869 711738 514268 779230 995561 272038 045825 > 2568 [i]