Best Known (28−12, 28, s)-Nets in Base 64
(28−12, 28, 685)-Net over F64 — Constructive and digital
Digital (16, 28, 685)-net over F64, using
- 641 times duplication [i] based on digital (15, 27, 685)-net over F64, using
- net defined by OOA [i] based on linear OOA(6427, 685, F64, 12, 12) (dual of [(685, 12), 8193, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6413, 4096, F64, 7) (dual of [4096, 4083, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(644, 14, F64, 4) (dual of [14, 10, 5]-code or 14-arc in PG(3,64)), using
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- Reed–Solomon code RS(60,64) [i]
- discarding factors / shortening the dual code based on linear OA(644, 64, F64, 4) (dual of [64, 60, 5]-code or 64-arc in PG(3,64)), using
- construction X applied to Ce(11) ⊂ Ce(6) [i] based on
- OA 6-folding and stacking [i] based on linear OA(6427, 4110, F64, 12) (dual of [4110, 4083, 13]-code), using
- net defined by OOA [i] based on linear OOA(6427, 685, F64, 12, 12) (dual of [(685, 12), 8193, 13]-NRT-code), using
(28−12, 28, 2731)-Net in Base 64 — Constructive
(16, 28, 2731)-net in base 64, using
- base change [i] based on digital (12, 24, 2731)-net over F128, using
- 1281 times duplication [i] based on digital (11, 23, 2731)-net over F128, using
- net defined by OOA [i] based on linear OOA(12823, 2731, F128, 12, 12) (dual of [(2731, 12), 32749, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12821, 16384, F128, 11) (dual of [16384, 16363, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(11) ⊂ Ce(10) [i] based on
- OA 6-folding and stacking [i] based on linear OA(12823, 16386, F128, 12) (dual of [16386, 16363, 13]-code), using
- net defined by OOA [i] based on linear OOA(12823, 2731, F128, 12, 12) (dual of [(2731, 12), 32749, 13]-NRT-code), using
- 1281 times duplication [i] based on digital (11, 23, 2731)-net over F128, using
(28−12, 28, 4344)-Net over F64 — Digital
Digital (16, 28, 4344)-net over F64, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(6428, 4344, F64, 12) (dual of [4344, 4316, 13]-code), using
- 241 step Varšamov–Edel lengthening with (ri) = (3, 7 times 0, 1, 44 times 0, 1, 187 times 0) [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(10) [i] based on
- linear OA(6423, 4096, F64, 12) (dual of [4096, 4073, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(6421, 4096, F64, 11) (dual of [4096, 4075, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(640, 2, F64, 0) (dual of [2, 2, 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(11) ⊂ Ce(10) [i] based on
- 241 step Varšamov–Edel lengthening with (ri) = (3, 7 times 0, 1, 44 times 0, 1, 187 times 0) [i] based on linear OA(6423, 4098, F64, 12) (dual of [4098, 4075, 13]-code), using
(28−12, 28, 5463)-Net in Base 64
(16, 28, 5463)-net in base 64, using
- base change [i] based on digital (12, 24, 5463)-net over F128, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12824, 5463, F128, 3, 12) (dual of [(5463, 3), 16365, 13]-NRT-code), using
- OOA 3-folding [i] based on linear OA(12824, 16389, F128, 12) (dual of [16389, 16365, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- linear OA(12823, 16384, F128, 12) (dual of [16384, 16361, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(12819, 16384, F128, 10) (dual of [16384, 16365, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 16383 = 1282−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(1281, 5, F128, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(1281, s, F128, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- OOA 3-folding [i] based on linear OA(12824, 16389, F128, 12) (dual of [16389, 16365, 13]-code), using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OOA(12824, 5463, F128, 3, 12) (dual of [(5463, 3), 16365, 13]-NRT-code), using
(28−12, 28, large)-Net in Base 64 — Upper bound on s
There is no (16, 28, large)-net in base 64, because
- 10 times m-reduction [i] would yield (16, 18, large)-net in base 64, but