Best Known (83−24, 83, s)-Nets in Base 32
(83−24, 83, 2774)-Net over F32 — Constructive and digital
Digital (59, 83, 2774)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 13, 44)-net over F32, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- Niederreiter–Xing sequence construction II/III [i] based on function field F/F32 with g(F) = 1 and N(F) ≥ 44, using
- net from sequence [i] based on digital (1, 43)-sequence over F32, using
- digital (46, 70, 2730)-net over F32, using
- net defined by OOA [i] based on linear OOA(3270, 2730, F32, 24, 24) (dual of [(2730, 24), 65450, 25]-NRT-code), using
- OA 12-folding and stacking [i] based on linear OA(3270, 32760, F32, 24) (dual of [32760, 32690, 25]-code), using
- discarding factors / shortening the dual code based on linear OA(3270, 32768, F32, 24) (dual of [32768, 32698, 25]-code), using
- an extension Ce(23) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,23], and designed minimum distance d ≥ |I|+1 = 24 [i]
- discarding factors / shortening the dual code based on linear OA(3270, 32768, F32, 24) (dual of [32768, 32698, 25]-code), using
- OA 12-folding and stacking [i] based on linear OA(3270, 32760, F32, 24) (dual of [32760, 32690, 25]-code), using
- net defined by OOA [i] based on linear OOA(3270, 2730, F32, 24, 24) (dual of [(2730, 24), 65450, 25]-NRT-code), using
- digital (1, 13, 44)-net over F32, using
(83−24, 83, 5462)-Net in Base 32 — Constructive
(59, 83, 5462)-net in base 32, using
- t-expansion [i] based on (58, 83, 5462)-net in base 32, using
- net defined by OOA [i] based on OOA(3283, 5462, S32, 25, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(3283, 65545, S32, 25), using
- 3 times code embedding in larger space [i] based on OA(3280, 65542, S32, 25), using
- discarding parts of the base [i] based on linear OA(25650, 65542, F256, 25) (dual of [65542, 65492, 26]-code), using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- linear OA(25649, 65537, F256, 25) (dual of [65537, 65488, 26]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,12], and minimum distance d ≥ |{−12,−11,…,12}|+1 = 26 (BCH-bound) [i]
- linear OA(25645, 65537, F256, 23) (dual of [65537, 65492, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,11], and minimum distance d ≥ |{−11,−10,…,11}|+1 = 24 (BCH-bound) [i]
- linear OA(2561, 5, F256, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(2561, s, F256, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to C([0,12]) ⊂ C([0,11]) [i] based on
- discarding parts of the base [i] based on linear OA(25650, 65542, F256, 25) (dual of [65542, 65492, 26]-code), using
- 3 times code embedding in larger space [i] based on OA(3280, 65542, S32, 25), using
- OOA 12-folding and stacking with additional row [i] based on OA(3283, 65545, S32, 25), using
- net defined by OOA [i] based on OOA(3283, 5462, S32, 25, 25), using
(83−24, 83, 82183)-Net over F32 — Digital
Digital (59, 83, 82183)-net over F32, using
(83−24, 83, large)-Net in Base 32 — Upper bound on s
There is no (59, 83, large)-net in base 32, because
- 22 times m-reduction [i] would yield (59, 61, large)-net in base 32, but