Best Known (109−33, 109, s)-Nets in Base 32
(109−33, 109, 2051)-Net over F32 — Constructive and digital
Digital (76, 109, 2051)-net over F32, using
- net defined by OOA [i] based on linear OOA(32109, 2051, F32, 33, 33) (dual of [(2051, 33), 67574, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(32109, 32817, F32, 33) (dual of [32817, 32708, 34]-code), using
- discarding factors / shortening the dual code based on linear OA(32109, 32819, F32, 33) (dual of [32819, 32710, 34]-code), using
- construction X applied to Ce(32) ⊂ Ce(19) [i] based on
- linear OA(3294, 32768, F32, 33) (dual of [32768, 32674, 34]-code), using an extension Ce(32) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,32], and designed minimum distance d ≥ |I|+1 = 33 [i]
- linear OA(3258, 32768, F32, 20) (dual of [32768, 32710, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(3215, 51, F32, 12) (dual of [51, 36, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3215, 63, F32, 12) (dual of [63, 48, 13]-code), using
- algebraic-geometric code AG(F,50P) [i] based on function field F/F32 with g(F) = 3 and N(F) ≥ 64, using
- discarding factors / shortening the dual code based on linear OA(3215, 63, F32, 12) (dual of [63, 48, 13]-code), using
- construction X applied to Ce(32) ⊂ Ce(19) [i] based on
- discarding factors / shortening the dual code based on linear OA(32109, 32819, F32, 33) (dual of [32819, 32710, 34]-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(32109, 32817, F32, 33) (dual of [32817, 32708, 34]-code), using
(109−33, 109, 4096)-Net in Base 32 — Constructive
(76, 109, 4096)-net in base 32, using
- 325 times duplication [i] based on (71, 104, 4096)-net in base 32, using
- base change [i] based on digital (32, 65, 4096)-net over F256, using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,16], and minimum distance d ≥ |{−16,−15,…,16}|+1 = 34 (BCH-bound) [i]
- OOA 16-folding and stacking with additional row [i] based on linear OA(25665, 65537, F256, 33) (dual of [65537, 65472, 34]-code), using
- net defined by OOA [i] based on linear OOA(25665, 4096, F256, 33, 33) (dual of [(4096, 33), 135103, 34]-NRT-code), using
- base change [i] based on digital (32, 65, 4096)-net over F256, using
(109−33, 109, 55279)-Net over F32 — Digital
Digital (76, 109, 55279)-net over F32, using
(109−33, 109, large)-Net in Base 32 — Upper bound on s
There is no (76, 109, large)-net in base 32, because
- 31 times m-reduction [i] would yield (76, 78, large)-net in base 32, but