Best Known (71, 71+18, s)-Nets in Base 64
(71, 71+18, 933093)-Net over F64 — Constructive and digital
Digital (71, 89, 933093)-net over F64, using
- (u, u+v)-construction [i] based on
- digital (11, 20, 1026)-net over F64, using
- net defined by OOA [i] based on linear OOA(6420, 1026, F64, 9, 9) (dual of [(1026, 9), 9214, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6420, 4105, F64, 9) (dual of [4105, 4085, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(6420, 4108, F64, 9) (dual of [4108, 4088, 10]-code), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- linear OA(6417, 4097, F64, 9) (dual of [4097, 4080, 10]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,4], and minimum distance d ≥ |{−4,−3,…,4}|+1 = 10 (BCH-bound) [i]
- linear OA(649, 4097, F64, 5) (dual of [4097, 4088, 6]-code), using the expurgated narrow-sense BCH-code C(I) with length 4097 | 644−1, defining interval I = [0,2], and minimum distance d ≥ |{−2,−1,0,1,2}|+1 = 6 (BCH-bound) [i]
- linear OA(643, 11, F64, 3) (dual of [11, 8, 4]-code or 11-arc in PG(2,64) or 11-cap in PG(2,64)), using
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- Reed–Solomon code RS(61,64) [i]
- discarding factors / shortening the dual code based on linear OA(643, 64, F64, 3) (dual of [64, 61, 4]-code or 64-arc in PG(2,64) or 64-cap in PG(2,64)), using
- construction X applied to C([0,4]) ⊂ C([0,2]) [i] based on
- discarding factors / shortening the dual code based on linear OA(6420, 4108, F64, 9) (dual of [4108, 4088, 10]-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(6420, 4105, F64, 9) (dual of [4105, 4085, 10]-code), using
- net defined by OOA [i] based on linear OOA(6420, 1026, F64, 9, 9) (dual of [(1026, 9), 9214, 10]-NRT-code), using
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- digital (11, 20, 1026)-net over F64, using
(71, 71+18, 936163)-Net in Base 64 — Constructive
(71, 89, 936163)-net in base 64, using
- (u, u+v)-construction [i] based on
- (11, 20, 4096)-net in base 64, using
- net defined by OOA [i] based on OOA(6420, 4096, S64, 9, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6420, 16385, S64, 9), using
- discarding factors based on OA(6420, 16386, S64, 9), using
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(7) [i] based on
- linear OA(12817, 16384, F128, 9) (dual of [16384, 16367, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(12815, 16384, F128, 8) (dual of [16384, 16369, 9]-code), using an extension Ce(7) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,7], and designed minimum distance d ≥ |I|+1 = 8 [i]
- linear OA(1280, 2, F128, 0) (dual of [2, 2, 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(8) ⊂ Ce(7) [i] based on
- discarding parts of the base [i] based on linear OA(12817, 16386, F128, 9) (dual of [16386, 16369, 10]-code), using
- discarding factors based on OA(6420, 16386, S64, 9), using
- OOA 4-folding and stacking with additional row [i] based on OA(6420, 16385, S64, 9), using
- net defined by OOA [i] based on OOA(6420, 4096, S64, 9, 9), using
- digital (51, 69, 932067)-net over F64, using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- OA 9-folding and stacking [i] based on linear OA(6469, large, F64, 18) (dual of [large, large−69, 19]-code), using
- net defined by OOA [i] based on linear OOA(6469, 932067, F64, 18, 18) (dual of [(932067, 18), 16777137, 19]-NRT-code), using
- (11, 20, 4096)-net in base 64, using
(71, 71+18, large)-Net over F64 — Digital
Digital (71, 89, large)-net over F64, using
- 641 times duplication [i] based on digital (70, 88, large)-net over F64, using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 16777215 = 644−1, defining interval I = [0,21], and designed minimum distance d ≥ |I|+1 = 23 [i]
- 3 times code embedding in larger space [i] based on linear OA(6485, large, F64, 22) (dual of [large, large−85, 23]-code), using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(6488, large, F64, 22) (dual of [large, large−88, 23]-code), using
- t-expansion [i] based on digital (66, 88, large)-net over F64, using
(71, 71+18, large)-Net in Base 64 — Upper bound on s
There is no (71, 89, large)-net in base 64, because
- 16 times m-reduction [i] would yield (71, 73, large)-net in base 64, but