Best Known (50, 62, s)-Nets in Base 5
(50, 62, 2615)-Net over F5 — Constructive and digital
Digital (50, 62, 2615)-net over F5, using
- (u, u+v)-construction [i] based on
- digital (1, 7, 10)-net over F5, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F5 with g(F) = 1 and N(F) ≥ 10, using
- net from sequence [i] based on digital (1, 9)-sequence over F5, using
- digital (43, 55, 2605)-net over F5, using
- net defined by OOA [i] based on linear OOA(555, 2605, F5, 12, 12) (dual of [(2605, 12), 31205, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(555, 15630, F5, 12) (dual of [15630, 15575, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(549, 15625, F5, 11) (dual of [15625, 15576, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(50, 6, F5, 0) (dual of [6, 6, 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(11) ⊂ Ce(10) [i] based on
- discarding factors / shortening the dual code based on linear OA(555, 15631, F5, 12) (dual of [15631, 15576, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(555, 15630, F5, 12) (dual of [15630, 15575, 13]-code), using
- net defined by OOA [i] based on linear OOA(555, 2605, F5, 12, 12) (dual of [(2605, 12), 31205, 13]-NRT-code), using
- digital (1, 7, 10)-net over F5, using
(50, 62, 15657)-Net over F5 — Digital
Digital (50, 62, 15657)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(562, 15657, F5, 12) (dual of [15657, 15595, 13]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(561, 15655, F5, 12) (dual of [15655, 15594, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(555, 15625, F5, 12) (dual of [15625, 15570, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(531, 15625, F5, 7) (dual of [15625, 15594, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(56, 30, F5, 4) (dual of [30, 24, 5]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(561, 15656, F5, 11) (dual of [15656, 15595, 12]-code), using Gilbert–Varšamov bound and bm = 561 > Vbs−1(k−1) = 254793 517487 716049 160540 092361 003309 491885 [i]
- linear OA(50, 1, F5, 0) (dual of [1, 1, 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(561, 15655, F5, 12) (dual of [15655, 15594, 13]-code), using
- construction X with Varšamov bound [i] based on
(50, 62, large)-Net in Base 5 — Upper bound on s
There is no (50, 62, large)-net in base 5, because
- 10 times m-reduction [i] would yield (50, 52, large)-net in base 5, but