Best Known (36, 36+9, s)-Nets in Base 64
(36, 36+9, 2228288)-Net over F64 — Constructive and digital
Digital (36, 45, 2228288)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (8, 12, 131138)-net over F64, using
- net defined by OOA [i] based on linear OOA(6412, 131138, F64, 4, 4) (dual of [(131138, 4), 524540, 5]-NRT-code), using
- appending kth column [i] based on linear OOA(6412, 131138, F64, 3, 4) (dual of [(131138, 3), 393402, 5]-NRT-code), using
- (u, u+v)-construction [i] based on
- linear OOA(642, 65, F64, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
- extended Reed–Solomon NRT-code RSe(3;193,64) [i]
- linear OOA(6410, 131073, F64, 3, 4) (dual of [(131073, 3), 393209, 5]-NRT-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- linear OA(6410, 262144, F64, 4) (dual of [262144, 262134, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(647, 262144, F64, 3) (dual of [262144, 262137, 4]-code or 262144-cap in PG(6,64)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [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(3) ⊂ Ce(2) [i] based on
- discarding factors / shortening the dual code based on linear OA(6410, 262147, F64, 4) (dual of [262147, 262137, 5]-code), using
- OA 2-folding and stacking [i] based on linear OA(6410, 262146, F64, 4) (dual of [262146, 262136, 5]-code), using
- linear OOA(642, 65, F64, 3, 2) (dual of [(65, 3), 193, 3]-NRT-code), using
- (u, u+v)-construction [i] based on
- appending kth column [i] based on linear OOA(6412, 131138, F64, 3, 4) (dual of [(131138, 3), 393402, 5]-NRT-code), using
- net defined by OOA [i] based on linear OOA(6412, 131138, F64, 4, 4) (dual of [(131138, 4), 524540, 5]-NRT-code), using
- digital (24, 33, 2097150)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 2097150, F64, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6433, 8388601, F64, 9) (dual of [8388601, 8388568, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6433, 8388601, F64, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(6433, 2097150, F64, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- digital (8, 12, 131138)-net over F64, using
(36, 36+9, 3145727)-Net in Base 64 — Constructive
(36, 45, 3145727)-net in base 64, using
- (u, u+v)-construction [i] based on
- (8, 12, 1048577)-net in base 64, using
- net defined by OOA [i] based on OOA(6412, 1048577, S64, 4, 4), using
- appending kth column [i] based on OOA(6412, 1048577, S64, 3, 4), using
- OA 2-folding and stacking [i] based on OA(6412, 2097154, S64, 4), using
- discarding factors based on OA(6412, 2097155, S64, 4), using
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- 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, 2097152, F128, 3) (dual of [2097152, 2097145, 4]-code or 2097152-cap in PG(6,128)), using an extension Ce(2) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 1283−1, defining interval I = [1,2], and designed minimum distance d ≥ |I|+1 = 3 [i]
- linear OA(1280, 3, F128, 0) (dual of [3, 3, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(1280, s, F128, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(3) ⊂ Ce(2) [i] based on
- discarding parts of the base [i] based on linear OA(12810, 2097155, F128, 4) (dual of [2097155, 2097145, 5]-code), using
- discarding factors based on OA(6412, 2097155, S64, 4), using
- OA 2-folding and stacking [i] based on OA(6412, 2097154, S64, 4), using
- appending kth column [i] based on OOA(6412, 1048577, S64, 3, 4), using
- net defined by OOA [i] based on OOA(6412, 1048577, S64, 4, 4), using
- digital (24, 33, 2097150)-net over F64, using
- net defined by OOA [i] based on linear OOA(6433, 2097150, F64, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6433, 8388601, F64, 9) (dual of [8388601, 8388568, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 16777217 | 648−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(6433, large, F64, 9) (dual of [large, large−33, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6433, 8388601, F64, 9) (dual of [8388601, 8388568, 10]-code), using
- net defined by OOA [i] based on linear OOA(6433, 2097150, F64, 9, 9) (dual of [(2097150, 9), 18874317, 10]-NRT-code), using
- (8, 12, 1048577)-net in base 64, using
(36, 36+9, large)-Net over F64 — Digital
Digital (36, 45, large)-net over F64, using
- t-expansion [i] based on digital (34, 45, large)-net over F64, using
- 1 times m-reduction [i] based on digital (34, 46, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6446, large, F64, 12) (dual of [large, large−46, 13]-code), using
- 1 times code embedding in larger space [i] 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]
- 1 times code embedding in larger space [i] based on linear OA(6445, large, F64, 12) (dual of [large, large−45, 13]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6446, large, F64, 12) (dual of [large, large−46, 13]-code), using
- 1 times m-reduction [i] based on digital (34, 46, large)-net over F64, using
(36, 36+9, large)-Net in Base 64 — Upper bound on s
There is no (36, 45, large)-net in base 64, because
- 7 times m-reduction [i] would yield (36, 38, large)-net in base 64, but