Best Known (26, 41, s)-Nets in Base 5
(26, 41, 132)-Net over F5 — Constructive and digital
Digital (26, 41, 132)-net over F5, using
- 3 times m-reduction [i] based on digital (26, 44, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 22, 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
- trace code for nets [i] based on digital (4, 22, 66)-net over F25, using
(26, 41, 176)-Net over F5 — Digital
Digital (26, 41, 176)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(541, 176, F5, 15) (dual of [176, 135, 16]-code), using
- 42 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 16 times 0) [i] based on linear OA(537, 130, F5, 15) (dual of [130, 93, 16]-code), using
- construction XX applied to C1 = C([123,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([123,13]) [i] based on
- linear OA(534, 124, F5, 14) (dual of [124, 90, 15]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−1,0,…,12}, and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(534, 124, F5, 14) (dual of [124, 90, 15]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,13], and designed minimum distance d ≥ |I|+1 = 15 [i]
- linear OA(537, 124, F5, 15) (dual of [124, 87, 16]-code), using the primitive BCH-code C(I) with length 124 = 53−1, defining interval I = {−1,0,…,13}, and designed minimum distance d ≥ |I|+1 = 16 [i]
- linear OA(531, 124, F5, 13) (dual of [124, 93, 14]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 124 = 53−1, defining interval I = [0,12], and designed minimum distance d ≥ |I|+1 = 14 [i]
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(50, s, F5, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(50, 3, F5, 0) (dual of [3, 3, 1]-code) (see above)
- construction XX applied to C1 = C([123,12]), C2 = C([0,13]), C3 = C1 + C2 = C([0,12]), and C∩ = C1 ∩ C2 = C([123,13]) [i] based on
- 42 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 1, 7 times 0, 1, 13 times 0, 1, 16 times 0) [i] based on linear OA(537, 130, F5, 15) (dual of [130, 93, 16]-code), using
(26, 41, 8331)-Net in Base 5 — Upper bound on s
There is no (26, 41, 8332)-net in base 5, because
- 1 times m-reduction [i] would yield (26, 40, 8332)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 9098 430369 946924 658135 038385 > 540 [i]