Best Known (66−10, 66, s)-Nets in Base 32
(66−10, 66, 2203066)-Net over F32 — Constructive and digital
Digital (56, 66, 2203066)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (15, 20, 525346)-net over F32, using
- net defined by OOA [i] based on linear OOA(3220, 525346, F32, 6, 5) (dual of [(525346, 6), 3152056, 6]-NRT-code), using
- OOA stacking with additional row [i] based on linear OOA(3220, 525347, F32, 2, 5) (dual of [(525347, 2), 1050674, 6]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(323, 1057, F32, 2, 2) (dual of [(1057, 2), 2111, 3]-NRT-code), using
- appending kth column [i] based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- Hamming code H(3,32) [i]
- appending kth column [i] based on linear OA(323, 1057, F32, 2) (dual of [1057, 1054, 3]-code), using
- linear OOA(3217, 524290, F32, 2, 5) (dual of [(524290, 2), 1048563, 6]-NRT-code), using
- OOA 2-folding [i] based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- construction X applied to Ce(4) ⊂ Ce(3) [i] based on
- linear OA(3217, 1048576, F32, 5) (dual of [1048576, 1048559, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 1048575 = 324−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [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(320, 4, F32, 0) (dual of [4, 4, 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(4) ⊂ Ce(3) [i] based on
- OOA 2-folding [i] based on linear OA(3217, 1048580, F32, 5) (dual of [1048580, 1048563, 6]-code), using
- linear OOA(323, 1057, F32, 2, 2) (dual of [(1057, 2), 2111, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- OOA stacking with additional row [i] based on linear OOA(3220, 525347, F32, 2, 5) (dual of [(525347, 2), 1050674, 6]-NRT-code), using
- net defined by OOA [i] based on linear OOA(3220, 525346, F32, 6, 5) (dual of [(525346, 6), 3152056, 6]-NRT-code), using
- digital (36, 46, 1677720)-net over F32, using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 33554431 = 325−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(3246, large, F32, 10) (dual of [large, large−46, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(3246, 8388600, F32, 10) (dual of [8388600, 8388554, 11]-code), using
- net defined by OOA [i] based on linear OOA(3246, 1677720, F32, 10, 10) (dual of [(1677720, 10), 16777154, 11]-NRT-code), using
- digital (15, 20, 525346)-net over F32, using
(66−10, 66, 3355440)-Net in Base 32 — Constructive
(56, 66, 3355440)-net in base 32, using
- base change [i] based on digital (45, 55, 3355440)-net over F64, using
- 641 times duplication [i] based on digital (44, 54, 3355440)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (12, 17, 4194301)-net over F64, using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- OOA 2-folding and stacking with additional row [i] based on linear OA(6417, large, F64, 5) (dual of [large, large−17, 6]-code), using
- net defined by OOA [i] based on linear OOA(6417, 4194301, F64, 5, 5) (dual of [(4194301, 5), 20971488, 6]-NRT-code), using
- digital (27, 37, 1677720)-net over F64, using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,9], and designed minimum distance d ≥ |I|+1 = 11 [i]
- discarding factors / shortening the dual code based on linear OA(6437, large, F64, 10) (dual of [large, large−37, 11]-code), using
- OA 5-folding and stacking [i] based on linear OA(6437, 8388600, F64, 10) (dual of [8388600, 8388563, 11]-code), using
- net defined by OOA [i] based on linear OOA(6437, 1677720, F64, 10, 10) (dual of [(1677720, 10), 16777163, 11]-NRT-code), using
- digital (12, 17, 4194301)-net over F64, using
- (u, u+v)-construction [i] based on
- 641 times duplication [i] based on digital (44, 54, 3355440)-net over F64, using
(66−10, 66, large)-Net over F32 — Digital
Digital (56, 66, large)-net over F32, using
- 5 times m-reduction [i] based on digital (56, 71, large)-net over F32, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,7], and minimum distance d ≥ |{−7,−6,…,7}|+1 = 16 (BCH-bound) [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(3271, large, F32, 15) (dual of [large, large−71, 16]-code), using
(66−10, 66, large)-Net in Base 32 — Upper bound on s
There is no (56, 66, large)-net in base 32, because
- 8 times m-reduction [i] would yield (56, 58, large)-net in base 32, but