Best Known (36, 36+33, s)-Nets in Base 25
(36, 36+33, 252)-Net over F25 — Constructive and digital
Digital (36, 69, 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, 43, 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
(36, 36+33, 589)-Net over F25 — Digital
Digital (36, 69, 589)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2569, 589, F25, 33) (dual of [589, 520, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(2569, 646, F25, 33) (dual of [646, 577, 34]-code), using
- construction XX applied to C1 = C([619,26]), C2 = C([3,27]), C3 = C1 + C2 = C([3,26]), and C∩ = C1 ∩ C2 = C([619,27]) [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 = {−5,−4,…,26}, and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(2549, 624, F25, 25) (dual of [624, 575, 26]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {3,4,…,27}, and designed minimum distance d ≥ |I|+1 = 26 [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 = {−5,−4,…,27}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(2547, 624, F25, 24) (dual of [624, 577, 25]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {3,4,…,26}, and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(257, 20, F25, 7) (dual of [20, 13, 8]-code or 20-arc in PG(6,25)), using
- discarding factors / shortening the dual code based on linear OA(257, 25, F25, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
- Reed–Solomon code RS(18,25) [i]
- discarding factors / shortening the dual code based on linear OA(257, 25, F25, 7) (dual of [25, 18, 8]-code or 25-arc in PG(6,25)), using
- 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
- construction XX applied to C1 = C([619,26]), C2 = C([3,27]), C3 = C1 + C2 = C([3,26]), and C∩ = C1 ∩ C2 = C([619,27]) [i] based on
- discarding factors / shortening the dual code based on linear OA(2569, 646, F25, 33) (dual of [646, 577, 34]-code), using
(36, 36+33, 247490)-Net in Base 25 — Upper bound on s
There is no (36, 69, 247491)-net in base 25, because
- 1 times m-reduction [i] would yield (36, 68, 247491)-net in base 25, but
- 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]