Best Known (41, 41+37, s)-Nets in Base 25
(41, 41+37, 288)-Net over F25 — Constructive and digital
Digital (41, 78, 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
(41, 41+37, 673)-Net over F25 — Digital
Digital (41, 78, 673)-net over F25, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(2578, 673, F25, 37) (dual of [673, 595, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2578, 679, F25, 37) (dual of [679, 601, 38]-code), using
- 43 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 23 times 0) [i] based on linear OA(2570, 628, F25, 37) (dual of [628, 558, 38]-code), using
- construction XX applied to C1 = C([623,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([623,35]) [i] based on
- linear OA(2568, 624, F25, 36) (dual of [624, 556, 37]-code), using the primitive BCH-code C(I) with length 624 = 252−1, defining interval I = {−1,0,…,34}, and designed minimum distance d ≥ |I|+1 = 37 [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(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(2566, 624, F25, 35) (dual of [624, 558, 36]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 252−1, defining interval I = [0,34], and designed minimum distance d ≥ |I|+1 = 36 [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([623,34]), C2 = C([0,35]), C3 = C1 + C2 = C([0,34]), and C∩ = C1 ∩ C2 = C([623,35]) [i] based on
- 43 step Varšamov–Edel lengthening with (ri) = (5, 0, 1, 4 times 0, 1, 11 times 0, 1, 23 times 0) [i] based on linear OA(2570, 628, F25, 37) (dual of [628, 558, 38]-code), using
- discarding factors / shortening the dual code based on linear OA(2578, 679, F25, 37) (dual of [679, 601, 38]-code), using
(41, 41+37, 300595)-Net in Base 25 — Upper bound on s
There is no (41, 78, 300596)-net in base 25, because
- 1 times m-reduction [i] would yield (41, 77, 300596)-net in base 25, but
- the generalized Rao bound for nets shows that 25m ≥ 437924 688611 053988 696338 732243 022636 605308 512803 977052 969257 582113 950004 254434 881161 059228 533905 825255 424065 > 2577 [i]