Best Known (52, 63, s)-Nets in Base 8
(52, 63, 104858)-Net over F8 — Constructive and digital
Digital (52, 63, 104858)-net over F8, using
- 81 times duplication [i] based on digital (51, 62, 104858)-net over F8, using
- net defined by OOA [i] based on linear OOA(862, 104858, F8, 11, 11) (dual of [(104858, 11), 1153376, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(862, 524291, F8, 11) (dual of [524291, 524229, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 524294, F8, 11) (dual of [524294, 524232, 12]-code), using
- trace code [i] based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(862, 524294, F8, 11) (dual of [524294, 524232, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(862, 524291, F8, 11) (dual of [524291, 524229, 12]-code), using
- net defined by OOA [i] based on linear OOA(862, 104858, F8, 11, 11) (dual of [(104858, 11), 1153376, 12]-NRT-code), using
(52, 63, 524296)-Net over F8 — Digital
Digital (52, 63, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(863, 524296, F8, 11) (dual of [524296, 524233, 12]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(862, 524294, F8, 11) (dual of [524294, 524232, 12]-code), using
- trace code [i] based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(6431, 262144, F64, 11) (dual of [262144, 262113, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [i]
- linear OA(6428, 262144, F64, 10) (dual of [262144, 262116, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [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(10) ⊂ Ce(9) [i] based on
- trace code [i] based on linear OA(6431, 262147, F64, 11) (dual of [262147, 262116, 12]-code), using
- linear OA(862, 524295, F8, 10) (dual of [524295, 524233, 11]-code), using Gilbert–Varšamov bound and bm = 862 > Vbs−1(k−1) = 332862 262700 076764 380298 269948 500697 233941 462263 201792 [i]
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(80, s, F8, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- linear OA(862, 524294, F8, 11) (dual of [524294, 524232, 12]-code), using
- construction X with Varšamov bound [i] based on
(52, 63, large)-Net in Base 8 — Upper bound on s
There is no (52, 63, large)-net in base 8, because
- 9 times m-reduction [i] would yield (52, 54, large)-net in base 8, but