Best Known (24−6, 24, s)-Nets in Base 8
(24−6, 24, 2734)-Net over F8 — Constructive and digital
Digital (18, 24, 2734)-net over F8, using
- net defined by OOA [i] based on linear OOA(824, 2734, F8, 6, 6) (dual of [(2734, 6), 16380, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(824, 8202, F8, 6) (dual of [8202, 8178, 7]-code), using
- trace code [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(824, 8202, F8, 6) (dual of [8202, 8178, 7]-code), using
(24−6, 24, 8202)-Net over F8 — Digital
Digital (18, 24, 8202)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(824, 8202, F8, 6) (dual of [8202, 8178, 7]-code), using
- trace code [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(6411, 4096, F64, 6) (dual of [4096, 4085, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(647, 4096, F64, 4) (dual of [4096, 4089, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 4095 = 642−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(641, 5, F64, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(641, s, F64, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- trace code [i] based on linear OA(6412, 4101, F64, 6) (dual of [4101, 4089, 7]-code), using
(24−6, 24, 4355173)-Net in Base 8 — Upper bound on s
There is no (18, 24, 4355174)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 4722 368023 834123 281830 > 824 [i]