Best Known (31, 42, s)-Nets in Base 16
(31, 42, 26215)-Net over F16 — Constructive and digital
Digital (31, 42, 26215)-net over F16, using
- net defined by OOA [i] based on linear OOA(1642, 26215, F16, 11, 11) (dual of [(26215, 11), 288323, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1642, 131076, F16, 11) (dual of [131076, 131034, 12]-code), using
- trace code [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(25621, 65536, F256, 11) (dual of [65536, 65515, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 65535 = 2562−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- 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(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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(25621, 65538, F256, 11) (dual of [65538, 65517, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(1642, 131076, F16, 11) (dual of [131076, 131034, 12]-code), using
(31, 42, 84542)-Net over F16 — Digital
Digital (31, 42, 84542)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1642, 84542, F16, 11) (dual of [84542, 84500, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(1642, 131074, F16, 11) (dual of [131074, 131032, 12]-code), using
- trace code [i] based on linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,5], and minimum distance d ≥ |{−5,−4,…,5}|+1 = 12 (BCH-bound) [i]
- trace code [i] based on linear OA(25621, 65537, F256, 11) (dual of [65537, 65516, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(1642, 131074, F16, 11) (dual of [131074, 131032, 12]-code), using
(31, 42, large)-Net in Base 16 — Upper bound on s
There is no (31, 42, large)-net in base 16, because
- 9 times m-reduction [i] would yield (31, 33, large)-net in base 16, but