Best Known (64, 64+21, s)-Nets in Base 5
(64, 64+21, 314)-Net over F5 — Constructive and digital
Digital (64, 85, 314)-net over F5, using
- net defined by OOA [i] based on linear OOA(585, 314, F5, 21, 21) (dual of [(314, 21), 6509, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(585, 3141, F5, 21) (dual of [3141, 3056, 22]-code), using
- discarding factors / shortening the dual code 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
- discarding factors / shortening the dual code based on linear OA(585, 3144, F5, 21) (dual of [3144, 3059, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(585, 3141, F5, 21) (dual of [3141, 3056, 22]-code), using
(64, 64+21, 2427)-Net over F5 — Digital
Digital (64, 85, 2427)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(585, 2427, F5, 21) (dual of [2427, 2342, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(585, 3140, F5, 21) (dual of [3140, 3055, 22]-code), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- linear OA(581, 3126, F5, 21) (dual of [3126, 3045, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 3126 | 510−1, defining interval I = [0,10], and minimum distance d ≥ |{−10,−9,…,10}|+1 = 22 (BCH-bound) [i]
- 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]
- linear OA(54, 14, F5, 3) (dual of [14, 10, 4]-code or 14-cap in PG(3,5)), using
- construction X applied to C([0,10]) ⊂ C([0,8]) [i] based on
- discarding factors / shortening the dual code based on linear OA(585, 3140, F5, 21) (dual of [3140, 3055, 22]-code), using
(64, 64+21, 841900)-Net in Base 5 — Upper bound on s
There is no (64, 85, 841901)-net in base 5, because
- 1 times m-reduction [i] would yield (64, 84, 841901)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 51699 131016 114464 064365 347318 723397 557284 703615 936477 499385 > 584 [i]