Best Known (89−16, 89, s)-Nets in Base 4
(89−16, 89, 2051)-Net over F4 — Constructive and digital
Digital (73, 89, 2051)-net over F4, using
- net defined by OOA [i] based on linear OOA(489, 2051, F4, 16, 16) (dual of [(2051, 16), 32727, 17]-NRT-code), using
- OA 8-folding and stacking [i] based on linear OA(489, 16408, F4, 16) (dual of [16408, 16319, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 16409, F4, 16) (dual of [16409, 16320, 17]-code), using
- 1 times truncation [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 16409, F4, 16) (dual of [16409, 16320, 17]-code), using
- OA 8-folding and stacking [i] based on linear OA(489, 16408, F4, 16) (dual of [16408, 16319, 17]-code), using
(89−16, 89, 12256)-Net over F4 — Digital
Digital (73, 89, 12256)-net over F4, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(489, 12256, F4, 16) (dual of [12256, 12167, 17]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 16409, F4, 16) (dual of [16409, 16320, 17]-code), using
- 1 times truncation [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- linear OA(485, 16384, F4, 17) (dual of [16384, 16299, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(464, 16384, F4, 13) (dual of [16384, 16320, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 16383 = 47−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(45, 26, F4, 3) (dual of [26, 21, 4]-code or 26-cap in PG(4,4)), using
- discarding factors / shortening the dual code based on linear OA(45, 41, F4, 3) (dual of [41, 36, 4]-code or 41-cap in PG(4,4)), using
- construction X applied to Ce(16) ⊂ Ce(12) [i] based on
- 1 times truncation [i] based on linear OA(490, 16410, F4, 17) (dual of [16410, 16320, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(489, 16409, F4, 16) (dual of [16409, 16320, 17]-code), using
(89−16, 89, 6258723)-Net in Base 4 — Upper bound on s
There is no (73, 89, 6258724)-net in base 4, because
- the generalized Rao bound for nets shows that 4m ≥ 383123 983572 595700 760796 498747 023969 869640 504491 402572 > 489 [i]