Best Known (71−22, 71, s)-Nets in Base 5
(71−22, 71, 252)-Net over F5 — Constructive and digital
Digital (49, 71, 252)-net over F5, using
- 7 times m-reduction [i] based on digital (49, 78, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- the Hermitian function field over F25 [i]
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 10 and N(F) ≥ 126, using
- net from sequence [i] based on digital (10, 125)-sequence over F25, using
- trace code for nets [i] based on digital (10, 39, 126)-net over F25, using
(71−22, 71, 567)-Net over F5 — Digital
Digital (49, 71, 567)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 567, F5, 22) (dual of [567, 496, 23]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 632, F5, 22) (dual of [632, 561, 23]-code), using
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- linear OA(569, 625, F5, 22) (dual of [625, 556, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(565, 625, F5, 21) (dual of [625, 560, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- linear OA(561, 625, F5, 19) (dual of [625, 564, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [i]
- linear OA(50, 5, F5, 0) (dual of [5, 5, 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(51, 2, F5, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(21) ⊂ Ce(20) ⊂ Ce(18) [i] based on
- discarding factors / shortening the dual code based on linear OA(571, 632, F5, 22) (dual of [632, 561, 23]-code), using
(71−22, 71, 39847)-Net in Base 5 — Upper bound on s
There is no (49, 71, 39848)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 42 355027 324336 020972 926452 189088 549245 013739 432353 > 571 [i]