Best Known (124, 124+21, s)-Nets in Base 5
(124, 124+21, 195312)-Net over F5 — Constructive and digital
Digital (124, 145, 195312)-net over F5, using
- net defined by OOA [i] based on linear OOA(5145, 195312, F5, 21, 21) (dual of [(195312, 21), 4101407, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5145, 1953121, F5, 21) (dual of [1953121, 1952976, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using
- an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1953124 = 59−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- discarding factors / shortening the dual code based on linear OA(5145, 1953125, F5, 21) (dual of [1953125, 1952980, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(5145, 1953121, F5, 21) (dual of [1953121, 1952976, 22]-code), using
(124, 124+21, 674513)-Net over F5 — Digital
Digital (124, 145, 674513)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(5145, 674513, F5, 2, 21) (dual of [(674513, 2), 1348881, 22]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(5145, 976563, F5, 2, 21) (dual of [(976563, 2), 1952981, 22]-NRT-code), using
- OOA 2-folding [i] based on linear OA(5145, 1953126, F5, 21) (dual of [1953126, 1952981, 22]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 1953126 | 518−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- OOA 2-folding [i] based on linear OA(5145, 1953126, F5, 21) (dual of [1953126, 1952981, 22]-code), using
- discarding factors / shortening the dual code based on linear OOA(5145, 976563, F5, 2, 21) (dual of [(976563, 2), 1952981, 22]-NRT-code), using
(124, 124+21, large)-Net in Base 5 — Upper bound on s
There is no (124, 145, large)-net in base 5, because
- 19 times m-reduction [i] would yield (124, 126, large)-net in base 5, but