Best Known (19, 19+15, s)-Nets in Base 25
(19, 19+15, 156)-Net over F25 — Constructive and digital
Digital (19, 34, 156)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (3, 10, 52)-net over F25, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 3 and N(F) ≥ 52, using
- net from sequence [i] based on digital (3, 51)-sequence over F25, using
- digital (9, 24, 104)-net over F25, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 9 and N(F) ≥ 104, using
- net from sequence [i] based on digital (9, 103)-sequence over F25, using
- digital (3, 10, 52)-net over F25, using
(19, 19+15, 704)-Net over F25 — Digital
Digital (19, 34, 704)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2534, 704, F25, 15) (dual of [704, 670, 16]-code), using
- 71 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 16 times 0, 1, 48 times 0) [i] based on linear OA(2529, 628, F25, 15) (dual of [628, 599, 16]-code), using
- construction XX applied to C1 = C([623,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([623,13]) [i] based on
- linear OA(2527, 624, F25, 14) (dual of [624, 597, 15]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2527, 624, F25, 14) (dual of [624, 597, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(2529, 624, F25, 15) (dual of [624, 595, 16]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(2525, 624, F25, 13) (dual of [624, 599, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [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([623,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([623,13]) [i] based on
- 71 step Varšamov–Edel lengthening with (ri) = (3, 4 times 0, 1, 16 times 0, 1, 48 times 0) [i] based on linear OA(2529, 628, F25, 15) (dual of [628, 599, 16]-code), using
(19, 19+15, 548268)-Net in Base 25 — Upper bound on s
There is no (19, 34, 548269)-net in base 25, because
- 1 times m-reduction [i] would yield (19, 33, 548269)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 13552 588231 343721 886280 517678 337607 290618 077001 > 2533 [i]