Best Known (54, 68, s)-Nets in Base 5
(54, 68, 2234)-Net over F5 — Constructive and digital
Digital (54, 68, 2234)-net over F5, using
- net defined by OOA [i] based on linear OOA(568, 2234, F5, 14, 14) (dual of [(2234, 14), 31208, 15]-NRT-code), using
- OA 7-folding and stacking [i] based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(51, 13, F5, 1) (dual of [13, 12, 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
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- OA 7-folding and stacking [i] based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
(54, 68, 10557)-Net over F5 — Digital
Digital (54, 68, 10557)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(568, 10557, F5, 14) (dual of [10557, 10489, 15]-code), using
- discarding factors / shortening the dual code based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- linear OA(567, 15625, F5, 14) (dual of [15625, 15558, 15]-code), using an extension Ce(13) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,13], and designed minimum distance d ≥ |I|+1 = 14 [i]
- 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(51, 13, F5, 1) (dual of [13, 12, 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
- construction X applied to Ce(13) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(568, 15638, F5, 14) (dual of [15638, 15570, 15]-code), using
(54, 68, 5210178)-Net in Base 5 — Upper bound on s
There is no (54, 68, 5210179)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 338813 481802 270477 532360 915948 242008 183014 827445 > 568 [i]