Best Known (117−20, 117, s)-Nets in Base 8
(117−20, 117, 52429)-Net over F8 — Constructive and digital
Digital (97, 117, 52429)-net over F8, using
- 81 times duplication [i] based on digital (96, 116, 52429)-net over F8, using
- net defined by OOA [i] based on linear OOA(8116, 52429, F8, 20, 20) (dual of [(52429, 20), 1048464, 21]-NRT-code), using
- OA 10-folding and stacking [i] based on linear OA(8116, 524290, F8, 20) (dual of [524290, 524174, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 524294, F8, 20) (dual of [524294, 524178, 21]-code), using
- trace code [i] based on linear OA(6458, 262147, F64, 20) (dual of [262147, 262089, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- trace code [i] based on linear OA(6458, 262147, F64, 20) (dual of [262147, 262089, 21]-code), using
- discarding factors / shortening the dual code based on linear OA(8116, 524294, F8, 20) (dual of [524294, 524178, 21]-code), using
- OA 10-folding and stacking [i] based on linear OA(8116, 524290, F8, 20) (dual of [524290, 524174, 21]-code), using
- net defined by OOA [i] based on linear OOA(8116, 52429, F8, 20, 20) (dual of [(52429, 20), 1048464, 21]-NRT-code), using
(117−20, 117, 524296)-Net over F8 — Digital
Digital (97, 117, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8117, 524296, F8, 20) (dual of [524296, 524179, 21]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8116, 524294, F8, 20) (dual of [524294, 524178, 21]-code), using
- trace code [i] based on linear OA(6458, 262147, F64, 20) (dual of [262147, 262089, 21]-code), using
- construction X applied to Ce(19) ⊂ Ce(18) [i] based on
- linear OA(6458, 262144, F64, 20) (dual of [262144, 262086, 21]-code), using an extension Ce(19) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,19], and designed minimum distance d ≥ |I|+1 = 20 [i]
- linear OA(6455, 262144, F64, 19) (dual of [262144, 262089, 20]-code), using an extension Ce(18) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,18], and designed minimum distance d ≥ |I|+1 = 19 [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(19) ⊂ Ce(18) [i] based on
- trace code [i] based on linear OA(6458, 262147, F64, 20) (dual of [262147, 262089, 21]-code), using
- linear OA(8116, 524295, F8, 19) (dual of [524295, 524179, 20]-code), using Gilbert–Varšamov bound and bm = 8116 > Vbs−1(k−1) = 2 278489 706442 357513 265334 843665 007490 405941 990034 511680 519431 099378 647624 821235 401301 666896 548908 171264 [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(8116, 524294, F8, 20) (dual of [524294, 524178, 21]-code), using
- construction X with Varšamov bound [i] based on
(117−20, 117, large)-Net in Base 8 — Upper bound on s
There is no (97, 117, large)-net in base 8, because
- 18 times m-reduction [i] would yield (97, 99, large)-net in base 8, but