Best Known (128−26, 128, s)-Nets in Base 16
(128−26, 128, 80662)-Net over F16 — Constructive and digital
Digital (102, 128, 80662)-net over F16, using
- 161 times duplication [i] based on digital (101, 127, 80662)-net over F16, using
- net defined by OOA [i] based on linear OOA(16127, 80662, F16, 26, 26) (dual of [(80662, 26), 2097085, 27]-NRT-code), using
- OA 13-folding and stacking [i] based on linear OA(16127, 1048606, F16, 26) (dual of [1048606, 1048479, 27]-code), using
- discarding factors / shortening the dual code based on linear OA(16127, 1048607, F16, 26) (dual of [1048607, 1048480, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(20) [i] based on
- linear OA(16121, 1048576, F16, 26) (dual of [1048576, 1048455, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1696, 1048576, F16, 21) (dual of [1048576, 1048480, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [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(25) ⊂ Ce(20) [i] based on
- discarding factors / shortening the dual code based on linear OA(16127, 1048607, F16, 26) (dual of [1048607, 1048480, 27]-code), using
- OA 13-folding and stacking [i] based on linear OA(16127, 1048606, F16, 26) (dual of [1048606, 1048479, 27]-code), using
- net defined by OOA [i] based on linear OOA(16127, 80662, F16, 26, 26) (dual of [(80662, 26), 2097085, 27]-NRT-code), using
(128−26, 128, 1048613)-Net over F16 — Digital
Digital (102, 128, 1048613)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(16128, 1048613, F16, 26) (dual of [1048613, 1048485, 27]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
- linear OA(16121, 1048576, F16, 26) (dual of [1048576, 1048455, 27]-code), using an extension Ce(25) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,25], and designed minimum distance d ≥ |I|+1 = 26 [i]
- linear OA(1691, 1048576, F16, 20) (dual of [1048576, 1048485, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 165−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(167, 37, F16, 5) (dual of [37, 30, 6]-code), using
- discarding factors / shortening the dual code based on linear OA(167, 241, F16, 5) (dual of [241, 234, 6]-code), using
- construction X applied to Ce(25) ⊂ Ce(19) [i] based on
(128−26, 128, large)-Net in Base 16 — Upper bound on s
There is no (102, 128, large)-net in base 16, because
- 24 times m-reduction [i] would yield (102, 104, large)-net in base 16, but