Best Known (38, 69, s)-Nets in Base 25
(38, 69, 252)-Net over F25 — Constructive and digital
Digital (38, 69, 252)-net over F25, using
- 5 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
(38, 69, 841)-Net over F25 — Digital
Digital (38, 69, 841)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2569, 841, F25, 31) (dual of [841, 772, 32]-code), using
- 202 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) [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
- 202 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) [i] based on linear OA(2558, 628, F25, 31) (dual of [628, 570, 32]-code), using
(38, 69, 581939)-Net in Base 25 — Upper bound on s
There is no (38, 69, 581940)-net in base 25, because
- 1 times m-reduction [i] would yield (38, 68, 581940)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 114796 201016 180820 262420 870084 036481 732364 444508 218722 515843 768491 680415 562292 475695 775435 839905 > 2568 [i]