Best Known (64, 64+20, s)-Nets in Base 5
(64, 64+20, 314)-Net over F5 — Constructive and digital
Digital (64, 84, 314)-net over F5, using
- net defined by OOA [i] based on linear OOA(584, 314, F5, 20, 20) (dual of [(314, 20), 6196, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(584, 3140, F5, 20) (dual of [3140, 3056, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 3143, F5, 20) (dual of [3143, 3059, 21]-code), using
- 1 times truncation [i] based on linear OA(585, 3144, F5, 21) (dual of [3144, 3059, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(585, 3144, F5, 21) (dual of [3144, 3059, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(584, 3143, F5, 20) (dual of [3143, 3059, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(584, 3140, F5, 20) (dual of [3140, 3056, 21]-code), using
(64, 64+20, 3143)-Net over F5 — Digital
Digital (64, 84, 3143)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(584, 3143, F5, 20) (dual of [3143, 3059, 21]-code), using
- 1 times truncation [i] based on linear OA(585, 3144, F5, 21) (dual of [3144, 3059, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- linear OA(581, 3125, F5, 21) (dual of [3125, 3044, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(54, 19, F5, 3) (dual of [19, 15, 4]-code or 19-cap in PG(3,5)), using
- construction X applied to Ce(20) ⊂ Ce(16) [i] based on
- 1 times truncation [i] based on linear OA(585, 3144, F5, 21) (dual of [3144, 3059, 22]-code), using
(64, 64+20, 841900)-Net in Base 5 — Upper bound on s
There is no (64, 84, 841901)-net in base 5, because
- the generalized Rao bound for nets shows that 5m ≥ 51699 131016 114464 064365 347318 723397 557284 703615 936477 499385 > 584 [i]