Best Known (133−28, 133, s)-Nets in Base 5
(133−28, 133, 1116)-Net over F5 — Constructive and digital
Digital (105, 133, 1116)-net over F5, using
- net defined by OOA [i] based on linear OOA(5133, 1116, F5, 28, 28) (dual of [(1116, 28), 31115, 29]-NRT-code), using
- OA 14-folding and stacking [i] based on linear OA(5133, 15624, F5, 28) (dual of [15624, 15491, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using
- OA 14-folding and stacking [i] based on linear OA(5133, 15624, F5, 28) (dual of [15624, 15491, 29]-code), using
(133−28, 133, 9311)-Net over F5 — Digital
Digital (105, 133, 9311)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(5133, 9311, F5, 28) (dual of [9311, 9178, 29]-code), using
- discarding factors / shortening the dual code based on linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using
- an extension Ce(27) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,27], and designed minimum distance d ≥ |I|+1 = 28 [i]
- discarding factors / shortening the dual code based on linear OA(5133, 15625, F5, 28) (dual of [15625, 15492, 29]-code), using
(133−28, 133, 6601036)-Net in Base 5 — Upper bound on s
There is no (105, 133, 6601037)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 918 356017 777724 405429 913426 683681 288559 829965 843680 736706 596291 530106 561886 645201 147448 766825 > 5133 [i]