Best Known (68, 68+13, s)-Nets in Base 32
(68, 68+13, 1410048)-Net over F32 — Constructive and digital
Digital (68, 81, 1410048)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (14, 20, 11948)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (1, 4, 1025)-net over F32, using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- appending kth column [i] based on linear OOA(324, 1025, F32, 2, 3) (dual of [(1025, 2), 2046, 4]-NRT-code), using
- net defined by OOA [i] based on linear OOA(324, 1025, F32, 3, 3) (dual of [(1025, 3), 3071, 4]-NRT-code), using
- digital (10, 16, 10923)-net over F32, using
- net defined by OOA [i] based on linear OOA(3216, 10923, F32, 6, 6) (dual of [(10923, 6), 65522, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(3216, 32769, F32, 6) (dual of [32769, 32753, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(3216, 32771, F32, 6) (dual of [32771, 32755, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(3216, 32768, F32, 6) (dual of [32768, 32752, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 323−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [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(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(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(3216, 32771, F32, 6) (dual of [32771, 32755, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(3216, 32769, F32, 6) (dual of [32769, 32753, 7]-code), using
- net defined by OOA [i] based on linear OOA(3216, 10923, F32, 6, 6) (dual of [(10923, 6), 65522, 7]-NRT-code), using
- digital (1, 4, 1025)-net over F32, using
- (u, u+v)-construction [i] based on
- digital (48, 61, 1398100)-net over F32, using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 33554433 | 3210−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(3261, large, F32, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(3261, 8388601, F32, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(3261, 1398100, F32, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (14, 20, 11948)-net over F32, using
(68, 68+13, 1747628)-Net in Base 32 — Constructive
(68, 81, 1747628)-net in base 32, 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
- (46, 59, 1398100)-net in base 32, using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- discarding factors based on OA(3259, large, S32, 13), using
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding parts of the base [i] based on linear OA(6449, large, F64, 13) (dual of [large, large−49, 14]-code), using
- discarding factors based on OA(3259, large, S32, 13), using
- OOA 6-folding and stacking with additional row [i] based on OA(3259, 8388601, S32, 13), using
- net defined by OOA [i] based on OOA(3259, 1398100, S32, 13, 13), using
- digital (16, 22, 349528)-net over F32, using
(68, 68+13, large)-Net over F32 — Digital
Digital (68, 81, large)-net over F32, using
- 5 times m-reduction [i] based on digital (68, 86, large)-net over F32, using
(68, 68+13, large)-Net in Base 32 — Upper bound on s
There is no (68, 81, large)-net in base 32, because
- 11 times m-reduction [i] would yield (68, 70, large)-net in base 32, but