Best Known (46, 67, s)-Nets in Base 5
(46, 67, 252)-Net over F5 — Constructive and digital
Digital (46, 67, 252)-net over F5, using
- 5 times m-reduction [i] based on digital (46, 72, 252)-net over F5, using
- trace code for nets [i] based on digital (10, 36, 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, 36, 126)-net over F25, using
(46, 67, 519)-Net over F5 — Digital
Digital (46, 67, 519)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(567, 519, F5, 21) (dual of [519, 452, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(567, 632, F5, 21) (dual of [632, 565, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- 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(557, 625, F5, 18) (dual of [625, 568, 19]-code), using an extension Ce(17) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,17], and designed minimum distance d ≥ |I|+1 = 18 [i]
- linear OA(51, 6, F5, 1) (dual of [6, 5, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(51, s, F5, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(20) ⊂ Ce(18) ⊂ Ce(17) [i] based on
- discarding factors / shortening the dual code based on linear OA(567, 632, F5, 21) (dual of [632, 565, 22]-code), using
(46, 67, 46457)-Net in Base 5 — Upper bound on s
There is no (46, 67, 46458)-net in base 5, because
- 1 times m-reduction [i] would yield (46, 66, 46458)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 13554 898149 741580 454279 783556 941544 893710 247745 > 566 [i]