Best Known (67, 67+17, s)-Nets in Base 5
(67, 67+17, 1956)-Net over F5 — Constructive and digital
Digital (67, 84, 1956)-net over F5, using
- net defined by OOA [i] based on linear OOA(584, 1956, F5, 17, 17) (dual of [(1956, 17), 33168, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(584, 15649, F5, 17) (dual of [15649, 15565, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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
- OOA 8-folding and stacking with additional row [i] based on linear OA(584, 15649, F5, 17) (dual of [15649, 15565, 18]-code), using
(67, 67+17, 11829)-Net over F5 — Digital
Digital (67, 84, 11829)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 11829, F5, 17) (dual of [11829, 11745, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 15649, F5, 17) (dual of [15649, 15565, 18]-code), using
- construction XX applied to Ce(16) ⊂ Ce(12) ⊂ Ce(11) [i] based on
- linear OA(579, 15625, F5, 17) (dual of [15625, 15546, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(561, 15625, F5, 13) (dual of [15625, 15564, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 15624 = 56−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [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(54, 23, F5, 3) (dual of [23, 19, 4]-code or 23-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(584, 15649, F5, 17) (dual of [15649, 15565, 18]-code), using
(67, 67+17, large)-Net in Base 5 — Upper bound on s
There is no (67, 84, large)-net in base 5, because
- 15 times m-reduction [i] would yield (67, 69, large)-net in base 5, but