Best Known (65, 65+21, s)-Nets in Base 5
(65, 65+21, 314)-Net over F5 — Constructive and digital
Digital (65, 86, 314)-net over F5, using
- 51 times duplication [i] based on 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
- net defined by OOA [i] based on linear OOA(585, 314, F5, 21, 21) (dual of [(314, 21), 6509, 22]-NRT-code), using
(65, 65+21, 2642)-Net over F5 — Digital
Digital (65, 86, 2642)-net over F5, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(586, 2642, F5, 21) (dual of [2642, 2556, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(586, 3146, F5, 21) (dual of [3146, 3060, 22]-code), using
- construction XX applied to Ce(20) ⊂ Ce(16) ⊂ Ce(15) [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(561, 3125, F5, 16) (dual of [3125, 3064, 17]-code), using an extension Ce(15) of the primitive narrow-sense BCH-code C(I) with length 3124 = 55−1, defining interval I = [1,15], and designed minimum distance d ≥ |I|+1 = 16 [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(20) ⊂ Ce(16) ⊂ Ce(15) [i] based on
- discarding factors / shortening the dual code based on linear OA(586, 3146, F5, 21) (dual of [3146, 3060, 22]-code), using
(65, 65+21, 988913)-Net in Base 5 — Upper bound on s
There is no (65, 86, 988914)-net in base 5, because
- 1 times m-reduction [i] would yield (65, 85, 988914)-net in base 5, but
- the generalized Rao bound for nets shows that 5m ≥ 258495 432557 045645 513437 964113 624721 465572 810886 165997 471681 > 585 [i]