Best Known (44, 44+15, s)-Nets in Base 5
(44, 44+15, 264)-Net over F5 — Constructive and digital
Digital (44, 59, 264)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (14, 21, 209)-net over F5, using
- net defined by OOA [i] based on linear OOA(521, 209, F5, 7, 7) (dual of [(209, 7), 1442, 8]-NRT-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- linear OA(521, 625, F5, 7) (dual of [625, 604, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(517, 625, F5, 6) (dual of [625, 608, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 624 = 54−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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
- construction X applied to Ce(6) ⊂ Ce(5) [i] based on
- discarding factors / shortening the dual code based on linear OA(521, 629, F5, 7) (dual of [629, 608, 8]-code), using
- OOA 3-folding and stacking with additional row [i] based on linear OA(521, 628, F5, 7) (dual of [628, 607, 8]-code), using
- net defined by OOA [i] based on linear OOA(521, 209, F5, 7, 7) (dual of [(209, 7), 1442, 8]-NRT-code), using
- digital (23, 38, 132)-net over F5, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F25 with g(F) = 4 and N(F) ≥ 66, using
- net from sequence [i] based on digital (4, 65)-sequence over F25, using
- trace code for nets [i] based on digital (4, 19, 66)-net over F25, using
- digital (14, 21, 209)-net over F5, using
(44, 44+15, 1562)-Net over F5 — Digital
Digital (44, 59, 1562)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(559, 1562, F5, 15) (dual of [1562, 1503, 16]-code), using
(44, 44+15, 522781)-Net in Base 5 — Upper bound on s
There is no (44, 59, 522782)-net in base 5, because
- 1 times m-reduction [i] would yield (44, 58, 522782)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 34694 561742 917305 378188 850303 781681 356185 > 558 [i]