Best Known (79−23, 79, s)-Nets in Base 32
(79−23, 79, 3023)-Net over F32 — Constructive and digital
Digital (56, 79, 3023)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 12, 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 (44, 67, 2979)-net over F32, using
- net defined by OOA [i] based on linear OOA(3267, 2979, F32, 23, 23) (dual of [(2979, 23), 68450, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3267, 32770, F32, 23) (dual of [32770, 32703, 24]-code), using
- discarding factors / shortening the dual code based on linear OA(3267, 32771, F32, 23) (dual of [32771, 32704, 24]-code), using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- linear OA(3267, 32768, F32, 23) (dual of [32768, 32701, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(3264, 32768, F32, 22) (dual of [32768, 32704, 23]-code), using an extension Ce(21) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,21], and designed minimum distance d ≥ |I|+1 = 22 [i]
- linear OA(320, 3, F32, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(320, s, F32, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(22) ⊂ Ce(21) [i] based on
- discarding factors / shortening the dual code based on linear OA(3267, 32771, F32, 23) (dual of [32771, 32704, 24]-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(3267, 32770, F32, 23) (dual of [32770, 32703, 24]-code), using
- net defined by OOA [i] based on linear OOA(3267, 2979, F32, 23, 23) (dual of [(2979, 23), 68450, 24]-NRT-code), using
- digital (1, 12, 44)-net over F32, using
(79−23, 79, 5959)-Net in Base 32 — Constructive
(56, 79, 5959)-net in base 32, using
- net defined by OOA [i] based on OOA(3279, 5959, S32, 23, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3279, 65550, S32, 23), using
- 2 times code embedding in larger space [i] based on OA(3277, 65548, S32, 23), using
- discarding parts of the base [i] based on linear OA(25648, 65548, F256, 23) (dual of [65548, 65500, 24]-code), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- 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(25637, 65537, F256, 19) (dual of [65537, 65500, 20]-code), using the expurgated narrow-sense BCH-code C(I) with length 65537 | 2564−1, defining interval I = [0,9], and minimum distance d ≥ |{−9,−8,…,9}|+1 = 20 (BCH-bound) [i]
- linear OA(2563, 11, F256, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,256) or 11-cap in PG(2,256)), using
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- Reed–Solomon code RS(253,256) [i]
- discarding factors / shortening the dual code based on linear OA(2563, 256, F256, 3) (dual of [256, 253, 4]-code or 256-arc in PG(2,256) or 256-cap in PG(2,256)), using
- construction X applied to C([0,11]) ⊂ C([0,9]) [i] based on
- discarding parts of the base [i] based on linear OA(25648, 65548, F256, 23) (dual of [65548, 65500, 24]-code), using
- 2 times code embedding in larger space [i] based on OA(3277, 65548, S32, 23), using
- OOA 11-folding and stacking with additional row [i] based on OA(3279, 65550, S32, 23), using
(79−23, 79, 74201)-Net over F32 — Digital
Digital (56, 79, 74201)-net over F32, using
(79−23, 79, large)-Net in Base 32 — Upper bound on s
There is no (56, 79, large)-net in base 32, because
- 21 times m-reduction [i] would yield (56, 58, large)-net in base 32, but