Best Known (35−9, 35, s)-Nets in Base 16
(35−9, 35, 32769)-Net over F16 — Constructive and digital
Digital (26, 35, 32769)-net over F16, using
- net defined by OOA [i] based on linear OOA(1635, 32769, F16, 9, 9) (dual of [(32769, 9), 294886, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1635, 131077, F16, 9) (dual of [131077, 131042, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1634, 131076, F16, 9) (dual of [131076, 131042, 10]-code), using
- trace code [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- 1 times code embedding in larger space [i] based on linear OA(1634, 131076, F16, 9) (dual of [131076, 131042, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(1635, 131077, F16, 9) (dual of [131077, 131042, 10]-code), using
(35−9, 35, 131078)-Net over F16 — Digital
Digital (26, 35, 131078)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1635, 131078, F16, 9) (dual of [131078, 131043, 10]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(1634, 131076, F16, 9) (dual of [131076, 131042, 10]-code), using
- trace code [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- 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(25615, 65536, F256, 8) (dual of [65536, 65521, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [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(8) ⊂ Ce(7) [i] based on
- trace code [i] based on linear OA(25617, 65538, F256, 9) (dual of [65538, 65521, 10]-code), using
- linear OA(1634, 131077, F16, 8) (dual of [131077, 131043, 9]-code), using Gilbert–Varšamov bound and bm = 1634 > Vbs−1(k−1) = 22532 143093 736542 987254 697443 790822 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(1634, 131076, F16, 9) (dual of [131076, 131042, 10]-code), using
- construction X with Varšamov bound [i] based on
(35−9, 35, large)-Net in Base 16 — Upper bound on s
There is no (26, 35, large)-net in base 16, because
- 7 times m-reduction [i] would yield (26, 28, large)-net in base 16, but