Best Known (54, 71, s)-Nets in Base 5
(54, 71, 393)-Net over F5 — Constructive and digital
Digital (54, 71, 393)-net over F5, using
- net defined by OOA [i] based on linear OOA(571, 393, F5, 17, 17) (dual of [(393, 17), 6610, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(571, 3145, F5, 17) (dual of [3145, 3074, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3146, F5, 17) (dual of [3146, 3075, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(566, 3125, F5, 17) (dual of [3125, 3059, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(551, 3125, F5, 13) (dual of [3125, 3074, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(546, 3125, F5, 12) (dual of [3125, 3079, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(54, 20, F5, 3) (dual of [20, 16, 4]-code or 20-cap in PG(3,5)), using
- linear OA(50, 1, F5, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- discarding factors / shortening the dual code based on linear OA(571, 3146, F5, 17) (dual of [3146, 3075, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(571, 3145, F5, 17) (dual of [3145, 3074, 18]-code), using
(54, 71, 2925)-Net over F5 — Digital
Digital (54, 71, 2925)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(571, 2925, F5, 17) (dual of [2925, 2854, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(571, 3126, F5, 17) (dual of [3126, 3055, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(571, 3126, F5, 17) (dual of [3126, 3055, 18]-code), using
(54, 71, 1229180)-Net in Base 5 — Upper bound on s
There is no (54, 71, 1229181)-net in base 5, because
- 1 times m-reduction [i] would yield (54, 70, 1229181)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 8 470355 960725 875028 098601 120791 342881 125753 280225 > 570 [i]