Best Known (72, 105, s)-Nets in Base 5
(72, 105, 296)-Net over F5 — Constructive and digital
Digital (72, 105, 296)-net over F5, using
- 1 times m-reduction [i] based on digital (72, 106, 296)-net over F5, using
- trace code for nets [i] based on digital (19, 53, 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, 53, 148)-net over F25, using
(72, 105, 652)-Net over F5 — Digital
Digital (72, 105, 652)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5105, 652, F5, 33) (dual of [652, 547, 34]-code), using
- 18 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 13 times 0) [i] based on linear OA(5103, 632, F5, 33) (dual of [632, 529, 34]-code), using
- construction XX applied to C1 = C([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- linear OA(599, 624, F5, 32) (dual of [624, 525, 33]-code), using the primitive BCH-code C(I) with length 624 = 54−1, defining interval I = {−1,0,…,30}, and designed minimum distance d ≥ |I|+1 = 33 [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(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(595, 624, F5, 31) (dual of [624, 529, 32]-code), using the primitive expurgated narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [0,30], and designed minimum distance d ≥ |I|+1 = 32 [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([623,30]), C2 = C([0,31]), C3 = C1 + C2 = C([0,30]), and C∩ = C1 ∩ C2 = C([623,31]) [i] based on
- 18 step Varšamov–Edel lengthening with (ri) = (1, 0, 0, 0, 1, 13 times 0) [i] based on linear OA(5103, 632, F5, 33) (dual of [632, 529, 34]-code), using
(72, 105, 59388)-Net in Base 5 — Upper bound on s
There is no (72, 105, 59389)-net in base 5, because
- 1 times m-reduction [i] would yield (72, 104, 59389)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 4 931531 176183 516511 459929 282124 325400 426652 724910 237458 919665 417298 610625 > 5104 [i]