Best Known (66, 78, s)-Nets in Base 32
(66, 78, 1747628)-Net over F32 — Constructive and digital
Digital (66, 78, 1747628)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (16, 22, 349528)-net over F32, using
- net defined by OOA [i] based on linear OOA(3222, 349528, F32, 6, 6) (dual of [(349528, 6), 2097146, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3222, 1048584, F32, 6) (dual of [1048584, 1048562, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(3222, 1048585, F32, 6) (dual of [1048585, 1048563, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(3221, 1048576, F32, 6) (dual of [1048576, 1048555, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(3213, 1048576, F32, 4) (dual of [1048576, 1048563, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(321, 9, F32, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(321, s, F32, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding factors / shortening the dual code based on linear OA(3222, 1048585, F32, 6) (dual of [1048585, 1048563, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(3222, 1048584, F32, 6) (dual of [1048584, 1048562, 7]-code), using
- net defined by OOA [i] based on linear OOA(3222, 349528, F32, 6, 6) (dual of [(349528, 6), 2097146, 7]-NRT-code), using
- digital (44, 56, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3256, 1398100, F32, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(3256, 8388600, F32, 12) (dual of [8388600, 8388544, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(3256, large, F32, 12) (dual of [large, large−56, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(3256, 8388600, F32, 12) (dual of [8388600, 8388544, 13]-code), using
- net defined by OOA [i] based on linear OOA(3256, 1398100, F32, 12, 12) (dual of [(1398100, 12), 16777144, 13]-NRT-code), using
- digital (16, 22, 349528)-net over F32, using
(66, 78, 2097153)-Net in Base 32 — Constructive
(66, 78, 2097153)-net in base 32, using
- (u, u+v)-construction [i] based on
- (18, 24, 699053)-net in base 32, using
- net defined by OOA [i] based on OOA(3224, 699053, S32, 6, 6), using
- OA 3-folding and stacking [i] based on OA(3224, 2097159, S32, 6), using
- discarding factors based on OA(3224, 2097160, S32, 6), using
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(12816, 2097152, F128, 6) (dual of [2097152, 2097136, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(12810, 2097152, F128, 4) (dual of [2097152, 2097142, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(1287, 8, F128, 7) (dual of [8, 1, 8]-code or 8-arc in PG(6,128)), using
- dual of repetition code with length 8 [i]
- linear OA(1281, 8, F128, 1) (dual of [8, 7, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- Reed–Solomon code RS(127,128) [i]
- discarding factors / shortening the dual code based on linear OA(1281, 128, F128, 1) (dual of [128, 127, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 2097160, F128, 6) (dual of [2097160, 2097143, 7]-code), using
- discarding factors based on OA(3224, 2097160, S32, 6), using
- OA 3-folding and stacking [i] based on OA(3224, 2097159, S32, 6), using
- net defined by OOA [i] based on OOA(3224, 699053, S32, 6, 6), using
- (42, 54, 1398100)-net in base 32, using
- base change [i] based on digital (33, 45, 1398100)-net over F64, using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,11], and designed minimum distance d ≥ |I|+1 = 13 [i]
- discarding factors / shortening the dual code based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(6445, 8388600, F64, 12) (dual of [8388600, 8388555, 13]-code), using
- net defined by OOA [i] based on linear OOA(6445, 1398100, F64, 12, 12) (dual of [(1398100, 12), 16777155, 13]-NRT-code), using
- base change [i] based on digital (33, 45, 1398100)-net over F64, using
- (18, 24, 699053)-net in base 32, using
(66, 78, large)-Net over F32 — Digital
Digital (66, 78, large)-net over F32, using
- t-expansion [i] based on digital (64, 78, large)-net over F32, using
- 3 times m-reduction [i] based on digital (64, 81, large)-net over F32, using
(66, 78, large)-Net in Base 32 — Upper bound on s
There is no (66, 78, large)-net in base 32, because
- 10 times m-reduction [i] would yield (66, 68, large)-net in base 32, but