Best Known (70−31, 70, s)-Nets in Base 25
(70−31, 70, 252)-Net over F25 — Constructive and digital
Digital (39, 70, 252)-net over F25, using
- 7 times m-reduction [i] based on digital (39, 77, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 29, 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, 48, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 29, 126)-net over F25, using
- (u, u+v)-construction [i] based on
(70−31, 70, 933)-Net over F25 — Digital
Digital (39, 70, 933)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2570, 933, F25, 31) (dual of [933, 863, 32]-code), using
- 293 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 55 times 0, 1, 74 times 0, 1, 90 times 0) [i] based on linear OA(2558, 628, F25, 31) (dual of [628, 570, 32]-code), using
- construction XX applied to C1 = C([623,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([623,29]) [i] based on
- linear OA(2556, 624, F25, 30) (dual of [624, 568, 31]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,28}, and designed minimum distance d ≥ |I|+1 = 31 [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(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(2554, 624, F25, 29) (dual of [624, 570, 30]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,28], and designed minimum distance d ≥ |I|+1 = 30 [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([623,28]), C2 = C([0,29]), C3 = C1 + C2 = C([0,28]), and C∩ = C1 ∩ C2 = C([623,29]) [i] based on
- 293 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 18 times 0, 1, 34 times 0, 1, 55 times 0, 1, 74 times 0, 1, 90 times 0) [i] based on linear OA(2558, 628, F25, 31) (dual of [628, 570, 32]-code), using
(70−31, 70, 721231)-Net in Base 25 — Upper bound on s
There is no (39, 70, 721232)-net in base 25, because
- 1 times m-reduction [i] would yield (39, 69, 721232)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2 869873 128892 543255 662816 743598 104047 699120 644361 926753 754323 166821 722230 979012 040121 808565 346945 > 2569 [i]