Best Known (86, 86+22, s)-Nets in Base 16
(86, 86+22, 95328)-Net over F16 — Constructive and digital
Digital (86, 108, 95328)-net over F16, using
- net defined by OOA [i] based on linear OOA(16108, 95328, F16, 22, 22) (dual of [(95328, 22), 2097108, 23]-NRT-code), using
- OA 11-folding and stacking [i] based on linear OA(16108, 1048608, F16, 22) (dual of [1048608, 1048500, 23]-code), using
- 1 times code embedding in larger space [i] based on linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- OA 11-folding and stacking [i] based on linear OA(16108, 1048608, F16, 22) (dual of [1048608, 1048500, 23]-code), using
(86, 86+22, 1048609)-Net over F16 — Digital
Digital (86, 108, 1048609)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16108, 1048609, F16, 22) (dual of [1048609, 1048501, 23]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16101, 1048576, F16, 22) (dual of [1048576, 1048475, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(1676, 1048576, F16, 17) (dual of [1048576, 1048500, 18]-code), using an extension Ce(16) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,16], and designed minimum distance d ≥ |I|+1 = 17 [i]
- linear OA(166, 31, F16, 4) (dual of [31, 25, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- 1 times truncation [i] based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(166, 240, F16, 4) (dual of [240, 234, 5]-code), using
- construction X applied to Ce(21) ⊂ Ce(16) [i] based on
- linear OA(16107, 1048608, F16, 21) (dual of [1048608, 1048501, 22]-code), using Gilbert–Varšamov bound and bm = 16107 > Vbs−1(k−1) = 353083 542769 778286 159053 903292 879633 438608 513276 807321 841008 434469 923162 797377 314897 661381 438096 266443 145068 147955 837072 963831 [i]
- linear OA(160, 1, F16, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(16107, 1048607, F16, 22) (dual of [1048607, 1048500, 23]-code), using
- construction X with Varšamov bound [i] based on
(86, 86+22, large)-Net in Base 16 — Upper bound on s
There is no (86, 108, large)-net in base 16, because
- 20 times m-reduction [i] would yield (86, 88, large)-net in base 16, but