Best Known (42−19, 42, s)-Nets in Base 25
(42−19, 42, 192)-Net over F25 — Constructive and digital
Digital (23, 42, 192)-net over F25, using
- (u, u+v)-construction [i] based on
- digital (4, 13, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- digital (10, 29, 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 (4, 13, 66)-net over F25, using
(42−19, 42, 668)-Net over F25 — Digital
Digital (23, 42, 668)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2542, 668, F25, 19) (dual of [668, 626, 20]-code), using
- 35 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
- construction XX applied to C1 = C([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- linear OA(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,16}, and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2535, 624, F25, 18) (dual of [624, 589, 19]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(2537, 624, F25, 19) (dual of [624, 587, 20]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,17}, and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(2533, 624, F25, 17) (dual of [624, 591, 18]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,16], and designed minimum distance d ≥ |I|+1 = 18 [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([623,16]), C2 = C([0,17]), C3 = C1 + C2 = C([0,16]), and C∩ = C1 ∩ C2 = C([623,17]) [i] based on
- 35 step Varšamov–Edel lengthening with (ri) = (3, 0, 0, 1, 7 times 0, 1, 23 times 0) [i] based on linear OA(2537, 628, F25, 19) (dual of [628, 591, 20]-code), using
(42−19, 42, 403579)-Net in Base 25 — Upper bound on s
There is no (23, 42, 403580)-net in base 25, because
- 1 times m-reduction [i] would yield (23, 41, 403580)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 2067 984302 331318 947538 534192 274743 736848 314281 910201 306785 > 2541 [i]