Best Known (42, 42+38, s)-Nets in Base 25
(42, 42+38, 288)-Net over F25 — Constructive and digital
Digital (42, 80, 288)-net over F25, using
- t-expansion [i] based on digital (41, 80, 288)-net over F25, using
- net from sequence [i] based on digital (41, 287)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 41 and N(F) ≥ 288, using
- net from sequence [i] based on digital (41, 287)-sequence over F25, using
(42, 42+38, 679)-Net over F25 — Digital
Digital (42, 80, 679)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2580, 679, F25, 38) (dual of [679, 599, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2580, 682, F25, 38) (dual of [682, 602, 39]-code), using
- 46 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0) [i] based on linear OA(2572, 628, F25, 38) (dual of [628, 556, 39]-code), using
- construction XX applied to C1 = C([623,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([623,36]) [i] based on
- linear OA(2570, 624, F25, 37) (dual of [624, 554, 38]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,35}, and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2570, 624, F25, 37) (dual of [624, 554, 38]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,36], and designed minimum distance d ≥ |I|+1 = 38 [i]
- linear OA(2572, 624, F25, 38) (dual of [624, 552, 39]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,36}, and designed minimum distance d ≥ |I|+1 = 39 [i]
- linear OA(2568, 624, F25, 36) (dual of [624, 556, 37]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,35], and designed minimum distance d ≥ |I|+1 = 37 [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([623,35]), C2 = C([0,36]), C3 = C1 + C2 = C([0,35]), and C∩ = C1 ∩ C2 = C([623,36]) [i] based on
- 46 step Varšamov–Edel lengthening with (ri) = (4, 1, 0, 1, 4 times 0, 1, 11 times 0, 1, 25 times 0) [i] based on linear OA(2572, 628, F25, 38) (dual of [628, 556, 39]-code), using
- discarding factors / shortening the dual code based on linear OA(2580, 682, F25, 38) (dual of [682, 602, 39]-code), using
(42, 42+38, 254126)-Net in Base 25 — Upper bound on s
There is no (42, 80, 254127)-net in base 25, because
- the generalized Rao bound for nets shows that 25m ≥ 6842 302704 216487 856023 125845 396838 352236 166714 192749 965059 963125 457528 576092 555037 471998 202692 354480 890131 293625 > 2580 [i]