Best Known (102, 102+21, s)-Nets in Base 8
(102, 102+21, 52429)-Net over F8 — Constructive and digital
Digital (102, 123, 52429)-net over F8, using
- 81 times duplication [i] based on digital (101, 122, 52429)-net over F8, using
- net defined by OOA [i] based on linear OOA(8122, 52429, F8, 21, 21) (dual of [(52429, 21), 1100887, 22]-NRT-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8122, 524291, F8, 21) (dual of [524291, 524169, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 524294, F8, 21) (dual of [524294, 524172, 22]-code), using
- trace code [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- discarding factors / shortening the dual code based on linear OA(8122, 524294, F8, 21) (dual of [524294, 524172, 22]-code), using
- OOA 10-folding and stacking with additional row [i] based on linear OA(8122, 524291, F8, 21) (dual of [524291, 524169, 22]-code), using
- net defined by OOA [i] based on linear OOA(8122, 52429, F8, 21, 21) (dual of [(52429, 21), 1100887, 22]-NRT-code), using
(102, 102+21, 524296)-Net over F8 — Digital
Digital (102, 123, 524296)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8123, 524296, F8, 21) (dual of [524296, 524173, 22]-code), using
- construction X with Varšamov bound [i] based on
- linear OA(8122, 524294, F8, 21) (dual of [524294, 524172, 22]-code), using
- trace code [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- construction X applied to Ce(20) ⊂ Ce(19) [i] based on
- linear OA(6461, 262144, F64, 21) (dual of [262144, 262083, 22]-code), using an extension Ce(20) of the primitive narrow-sense BCH-code C(I) with length 262143 = 643−1, defining interval I = [1,20], and designed minimum distance d ≥ |I|+1 = 21 [i]
- 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(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(20) ⊂ Ce(19) [i] based on
- trace code [i] based on linear OA(6461, 262147, F64, 21) (dual of [262147, 262086, 22]-code), using
- linear OA(8122, 524295, F8, 20) (dual of [524295, 524173, 21]-code), using Gilbert–Varšamov bound and bm = 8122 > Vbs−1(k−1) = 440100 240206 448607 498792 836001 080175 872434 311946 680993 171096 544036 275548 479054 611874 954666 715087 871828 099072 [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(8122, 524294, F8, 21) (dual of [524294, 524172, 22]-code), using
- construction X with Varšamov bound [i] based on
(102, 102+21, large)-Net in Base 8 — Upper bound on s
There is no (102, 123, large)-net in base 8, because
- 19 times m-reduction [i] would yield (102, 104, large)-net in base 8, but