Best Known (88, 88+17, s)-Nets in Base 32
(88, 88+17, 1056769)-Net over F32 — Constructive and digital
Digital (88, 105, 1056769)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (16, 24, 8194)-net over F32, using
- net defined by OOA [i] based on linear OOA(3224, 8194, F32, 8, 8) (dual of [(8194, 8), 65528, 9]-NRT-code), using
- OA 4-folding and stacking [i] based on linear OA(3224, 32776, F32, 8) (dual of [32776, 32752, 9]-code), using
- discarding factors / shortening the dual code based on linear OA(3224, 32779, F32, 8) (dual of [32779, 32755, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(4) [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(3213, 32768, F32, 5) (dual of [32768, 32755, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(322, 11, F32, 2) (dual of [11, 9, 3]-code or 11-arc in PG(1,32)), using
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- Reed–Solomon code RS(30,32) [i]
- discarding factors / shortening the dual code based on linear OA(322, 32, F32, 2) (dual of [32, 30, 3]-code or 32-arc in PG(1,32)), using
- construction X applied to Ce(7) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(3224, 32779, F32, 8) (dual of [32779, 32755, 9]-code), using
- OA 4-folding and stacking [i] based on linear OA(3224, 32776, F32, 8) (dual of [32776, 32752, 9]-code), using
- net defined by OOA [i] based on linear OOA(3224, 8194, F32, 8, 8) (dual of [(8194, 8), 65528, 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 (16, 24, 8194)-net over F32, using
(88, 88+17, 1114111)-Net in Base 32 — Constructive
(88, 105, 1114111)-net in base 32, using
- (u, u+v)-construction [i] based on
- (19, 27, 65536)-net in base 32, using
- net defined by OOA [i] based on OOA(3227, 65536, S32, 8, 8), using
- OA 4-folding and stacking [i] based on OA(3227, 262144, S32, 8), using
- discarding factors based on OA(3227, 262147, S32, 8), using
- discarding parts of the base [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- construction X applied to Ce(7) ⊂ Ce(6) [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(6419, 262144, F64, 7) (dual of [262144, 262125, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(640, 3, F64, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(640, s, F64, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(7) ⊂ Ce(6) [i] based on
- discarding parts of the base [i] based on linear OA(6422, 262147, F64, 8) (dual of [262147, 262125, 9]-code), using
- discarding factors based on OA(3227, 262147, S32, 8), using
- OA 4-folding and stacking [i] based on OA(3227, 262144, S32, 8), using
- net defined by OOA [i] based on OOA(3227, 65536, 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
- (19, 27, 65536)-net in base 32, using
(88, 88+17, large)-Net over F32 — Digital
Digital (88, 105, large)-net over F32, using
- t-expansion [i] based on digital (87, 105, large)-net over F32, using
- 5 times m-reduction [i] based on digital (87, 110, large)-net over F32, using
(88, 88+17, large)-Net in Base 32 — Upper bound on s
There is no (88, 105, large)-net in base 32, because
- 15 times m-reduction [i] would yield (88, 90, large)-net in base 32, but