Best Known (39−10, 39, s)-Nets in Base 16
(39−10, 39, 26215)-Net over F16 — Constructive and digital
Digital (29, 39, 26215)-net over F16, using
- 161 times duplication [i] based on digital (28, 38, 26215)-net over F16, using
- net defined by OOA [i] based on linear OOA(1638, 26215, F16, 10, 10) (dual of [(26215, 10), 262112, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(1638, 131075, F16, 10) (dual of [131075, 131037, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(1638, 131075, F16, 10) (dual of [131075, 131037, 11]-code), using
- net defined by OOA [i] based on linear OOA(1638, 26215, F16, 10, 10) (dual of [(26215, 10), 262112, 11]-NRT-code), using
(39−10, 39, 131078)-Net over F16 — Digital
Digital (29, 39, 131078)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1639, 131078, F16, 10) (dual of [131078, 131039, 11]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- linear OA(25619, 65536, F256, 10) (dual of [65536, 65517, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(25617, 65536, F256, 9) (dual of [65536, 65519, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(2560, 2, F256, 0) (dual of [2, 2, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(2560, s, F256, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(9) ⊂ Ce(8) [i] based on
- trace code [i] based on linear OA(25619, 65538, F256, 10) (dual of [65538, 65519, 11]-code), using
- linear OA(1638, 131077, F16, 9) (dual of [131077, 131039, 10]-code), using Gilbert–Varšamov bound and bm = 1638 > Vbs−1(k−1) = 5537 375560 090126 278531 029695 412988 910822 998016 [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(1638, 131076, F16, 10) (dual of [131076, 131038, 11]-code), using
- construction X with Varšamov bound [i] based on
(39−10, 39, large)-Net in Base 16 — Upper bound on s
There is no (29, 39, large)-net in base 16, because
- 8 times m-reduction [i] would yield (29, 31, large)-net in base 16, but