Best Known (106−17, 106, s)-Nets in Base 32
(106−17, 106, 1056770)-Net over F32 — Constructive and digital
Digital (89, 106, 1056770)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (17, 25, 8195)-net over F32, using
- net defined by OOA [i] based on linear OOA(3225, 8195, F32, 8, 8) (dual of [(8195, 8), 65535, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3225, 32780, F32, 8) (dual of [32780, 32755, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3225, 32783, F32, 8) (dual of [32783, 32758, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- linear OA(3222, 32768, F32, 8) (dual of [32768, 32746, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(3210, 32768, F32, 4) (dual of [32768, 32758, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(323, 15, F32, 3) (dual of [15, 12, 4]-code or 15-arc in PG(2,32) or 15-cap in PG(2,32)), using
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- Reed–Solomon code RS(29,32) [i]
- discarding factors / shortening the dual code based on linear OA(323, 32, F32, 3) (dual of [32, 29, 4]-code or 32-arc in PG(2,32) or 32-cap in PG(2,32)), using
- construction X applied to Ce(7) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3225, 32783, F32, 8) (dual of [32783, 32758, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(3225, 32780, F32, 8) (dual of [32780, 32755, 9]-code), using
- net defined by OOA [i] based on linear OOA(3225, 8195, F32, 8, 8) (dual of [(8195, 8), 65535, 9]-NRT-code), using
- digital (64, 81, 1048575)-net over F32, using
- net defined by OOA [i] based on linear OOA(3281, 1048575, F32, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3281, 8388601, F32, 17) (dual of [8388601, 8388520, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(3281, large, F32, 17) (dual of [large, large−81, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3281, large, F32, 17) (dual of [large, large−81, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(3281, 8388601, F32, 17) (dual of [8388601, 8388520, 18]-code), using
- net defined by OOA [i] based on linear OOA(3281, 1048575, F32, 17, 17) (dual of [(1048575, 17), 17825694, 18]-NRT-code), using
- digital (17, 25, 8195)-net over F32, using
(106−17, 106, 1114112)-Net in Base 32 — Constructive
(89, 106, 1114112)-net in base 32, using
- (u, u+v)-construction [i] based on
- (20, 28, 65537)-net in base 32, using
- net defined by OOA [i] based on OOA(3228, 65537, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3228, 262148, S32, 8), using
- discarding factors based on OA(3228, 262151, S32, 8), using
- discarding parts of the base [i] based on linear OA(6423, 262151, F64, 8) (dual of [262151, 262128, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- linear OA(6422, 262144, F64, 8) (dual of [262144, 262122, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(6416, 262144, F64, 6) (dual of [262144, 262128, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(641, 7, F64, 1) (dual of [7, 6, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(5) [i] based on
- discarding parts of the base [i] based on linear OA(6423, 262151, F64, 8) (dual of [262151, 262128, 9]-code), using
- discarding factors based on OA(3228, 262151, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3228, 262148, S32, 8), using
- net defined by OOA [i] based on OOA(3228, 65537, S32, 8, 8), using
- (61, 78, 1048575)-net in base 32, using
- base change [i] based on digital (48, 65, 1048575)-net over F64, using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,8], and minimum distance d ≥ |{−8,−7,…,8}|+1 = 18 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6465, large, F64, 17) (dual of [large, large−65, 18]-code), using
- OOA 8-folding and stacking with additional row [i] based on linear OA(6465, 8388601, F64, 17) (dual of [8388601, 8388536, 18]-code), using
- net defined by OOA [i] based on linear OOA(6465, 1048575, F64, 17, 17) (dual of [(1048575, 17), 17825710, 18]-NRT-code), using
- base change [i] based on digital (48, 65, 1048575)-net over F64, using
- (20, 28, 65537)-net in base 32, using
(106−17, 106, large)-Net over F32 — Digital
Digital (89, 106, large)-net over F32, using
- t-expansion [i] based on digital (87, 106, large)-net over F32, using
- 4 times m-reduction [i] based on digital (87, 110, large)-net over F32, using
(106−17, 106, large)-Net in Base 32 — Upper bound on s
There is no (89, 106, large)-net in base 32, because
- 15 times m-reduction [i] would yield (89, 91, large)-net in base 32, but