Best Known (12, 12+13, s)-Nets in Base 25
(12, 12+13, 126)-Net over F25 — Constructive and digital
Digital (12, 25, 126)-net over F25, using
- t-expansion [i] based on digital (10, 25, 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
(12, 12+13, 305)-Net over F25 — Digital
Digital (12, 25, 305)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(2525, 305, F25, 2, 13) (dual of [(305, 2), 585, 14]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2525, 314, F25, 2, 13) (dual of [(314, 2), 603, 14]-NRT-code), using
- OOA 2-folding [i] based on linear OA(2525, 628, F25, 13) (dual of [628, 603, 14]-code), using
- construction XX applied to C1 = C([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- linear OA(2523, 624, F25, 12) (dual of [624, 601, 13]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,10}, and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2523, 624, F25, 12) (dual of [624, 601, 13]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(2525, 624, F25, 13) (dual of [624, 599, 14]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,11}, and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(2521, 624, F25, 11) (dual of [624, 603, 12]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,10], and designed minimum distance d ≥ |I|+1 = 12 [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([623,10]), C2 = C([0,11]), C3 = C1 + C2 = C([0,10]), and C∩ = C1 ∩ C2 = C([623,11]) [i] based on
- OOA 2-folding [i] based on linear OA(2525, 628, F25, 13) (dual of [628, 603, 14]-code), using
- discarding factors / shortening the dual code based on linear OOA(2525, 314, F25, 2, 13) (dual of [(314, 2), 603, 14]-NRT-code), using
(12, 12+13, 48724)-Net in Base 25 — Upper bound on s
There is no (12, 25, 48725)-net in base 25, because
- 1 times m-reduction [i] would yield (12, 24, 48725)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 3552 982770 838550 931038 241164 555281 > 2524 [i]