Best Known (21, 21+9, s)-Nets in Base 8
(21, 21+9, 1025)-Net over F8 — Constructive and digital
Digital (21, 30, 1025)-net over F8, using
- net defined by OOA [i] based on linear OOA(830, 1025, F8, 9, 9) (dual of [(1025, 9), 9195, 10]-NRT-code), using
- OOA 4-folding and stacking with additional row [i] based on linear OA(830, 4101, F8, 9) (dual of [4101, 4071, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(825, 4096, F8, 7) (dual of [4096, 4071, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 5, F8, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- OOA 4-folding and stacking with additional row [i] based on linear OA(830, 4101, F8, 9) (dual of [4101, 4071, 10]-code), using
(21, 21+9, 2658)-Net over F8 — Digital
Digital (21, 30, 2658)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(830, 2658, F8, 9) (dual of [2658, 2628, 10]-code), using
- discarding factors / shortening the dual code based on linear OA(830, 4101, F8, 9) (dual of [4101, 4071, 10]-code), using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- linear OA(829, 4096, F8, 9) (dual of [4096, 4067, 10]-code), using an extension Ce(8) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,8], and designed minimum distance d ≥ |I|+1 = 9 [i]
- linear OA(825, 4096, F8, 7) (dual of [4096, 4071, 8]-code), using an extension Ce(6) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,6], and designed minimum distance d ≥ |I|+1 = 7 [i]
- linear OA(81, 5, F8, 1) (dual of [5, 4, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, s, F8, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(8) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(830, 4101, F8, 9) (dual of [4101, 4071, 10]-code), using
(21, 21+9, 1115209)-Net in Base 8 — Upper bound on s
There is no (21, 30, 1115210)-net in base 8, because
- 1 times m-reduction [i] would yield (21, 29, 1115210)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 154 742911 989078 862916 106606 > 829 [i]