Best Known (75, 109, s)-Nets in Base 5
(75, 109, 296)-Net over F5 — Constructive and digital
Digital (75, 109, 296)-net over F5, using
- 3 times m-reduction [i] based on digital (75, 112, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 56, 148)-net over F25, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 19 and N(F) ≥ 148, using
- net from sequence [i] based on digital (19, 147)-sequence over F25, using
- trace code for nets [i] based on digital (19, 56, 148)-net over F25, using
(75, 109, 687)-Net over F5 — Digital
Digital (75, 109, 687)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5109, 687, F5, 34) (dual of [687, 578, 35]-code), using
- 53 step Varšamov–Edel lengthening with (ri) = (1, 22 times 0, 1, 29 times 0) [i] based on linear OA(5107, 632, F5, 34) (dual of [632, 525, 35]-code), using
- construction XX applied to C1 = C([623,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([623,32]) [i] based on
- linear OA(5103, 624, F5, 33) (dual of [624, 521, 34]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,31}, and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(5103, 624, F5, 33) (dual of [624, 521, 34]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,32], and designed minimum distance d ≥ |I|+1 = 34 [i]
- linear OA(5107, 624, F5, 34) (dual of [624, 517, 35]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,32}, and designed minimum distance d ≥ |I|+1 = 35 [i]
- linear OA(599, 624, F5, 32) (dual of [624, 525, 33]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,31], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(50, 4, F5, 0) (dual of [4, 4, 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, 4, F5, 0) (dual of [4, 4, 1]-code) (see above)
- construction XX applied to C1 = C([623,31]), C2 = C([0,32]), C3 = C1 + C2 = C([0,31]), and C∩ = C1 ∩ C2 = C([623,32]) [i] based on
- 53 step Varšamov–Edel lengthening with (ri) = (1, 22 times 0, 1, 29 times 0) [i] based on linear OA(5107, 632, F5, 34) (dual of [632, 525, 35]-code), using
(75, 109, 54377)-Net in Base 5 — Upper bound on s
There is no (75, 109, 54378)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 15410 230568 242857 678329 807005 563493 477329 266487 484586 901144 463672 961618 547625 > 5109 [i]