Best Known (71−33, 71, s)-Nets in Base 25
(71−33, 71, 252)-Net over F25 — Constructive and digital
Digital (38, 71, 252)-net over F25, using
- 3 times m-reduction [i] based on digital (38, 74, 252)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (10, 28, 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, 46, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25 (see above)
- digital (10, 28, 126)-net over F25, using
- (u, u+v)-construction [i] based on
(71−33, 71, 711)-Net over F25 — Digital
Digital (38, 71, 711)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2571, 711, F25, 33) (dual of [711, 640, 34]-code), using
- 74 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 20 times 0, 1, 35 times 0) [i] based on linear OA(2562, 628, F25, 33) (dual of [628, 566, 34]-code), using
- construction XX applied to C1 = C([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- 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(2560, 624, F25, 32) (dual of [624, 564, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2562, 624, F25, 33) (dual of [624, 562, 34]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [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(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([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- 74 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 9 times 0, 1, 20 times 0, 1, 35 times 0) [i] based on linear OA(2562, 628, F25, 33) (dual of [628, 566, 34]-code), using
(71−33, 71, 370088)-Net in Base 25 — Upper bound on s
There is no (38, 71, 370089)-net in base 25, because
- 1 times m-reduction [i] would yield (38, 70, 370089)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 71 748435 843417 893032 012117 367553 068524 622201 687563 864131 446476 570416 551731 498043 938780 868296 635265 > 2570 [i]