Best Known (22, 28, s)-Nets in Base 8
(22, 28, 10927)-Net over F8 — Constructive and digital
Digital (22, 28, 10927)-net over F8, using
- net defined by OOA [i] based on linear OOA(828, 10927, F8, 6, 6) (dual of [(10927, 6), 65534, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(828, 32781, F8, 6) (dual of [32781, 32753, 7]-code), using
- 1 times code embedding in larger space [i] based on linear OA(827, 32780, F8, 6) (dual of [32780, 32753, 7]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(826, 32768, F8, 6) (dual of [32768, 32742, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(816, 32768, F8, 4) (dual of [32768, 32752, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 32767 = 85−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(811, 12, F8, 11) (dual of [12, 1, 12]-code or 12-arc in PG(10,8)), using
- dual of repetition code with length 12 [i]
- linear OA(81, 12, F8, 1) (dual of [12, 11, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 511 = 83−1, defining interval I = [0,0], and designed minimum distance d ≥ |I|+1 = 2 [i]
- discarding factors / shortening the dual code based on linear OA(81, 511, F8, 1) (dual of [511, 510, 2]-code), using
- construction X4 applied to Ce(5) ⊂ Ce(3) [i] based on
- 1 times code embedding in larger space [i] based on linear OA(827, 32780, F8, 6) (dual of [32780, 32753, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(828, 32781, F8, 6) (dual of [32781, 32753, 7]-code), using
(22, 28, 21846)-Net in Base 8 — Constructive
(22, 28, 21846)-net in base 8, using
- base change [i] based on digital (15, 21, 21846)-net over F16, using
- net defined by OOA [i] based on linear OOA(1621, 21846, F16, 6, 6) (dual of [(21846, 6), 131055, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(1621, 65538, F16, 6) (dual of [65538, 65517, 7]-code), using
- discarding factors / shortening the dual code based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- discarding factors / shortening the dual code based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- OA 3-folding and stacking [i] based on linear OA(1621, 65538, F16, 6) (dual of [65538, 65517, 7]-code), using
- net defined by OOA [i] based on linear OOA(1621, 21846, F16, 6, 6) (dual of [(21846, 6), 131055, 7]-NRT-code), using
(22, 28, 42468)-Net over F8 — Digital
Digital (22, 28, 42468)-net over F8, using
(22, 28, 65540)-Net in Base 8
(22, 28, 65540)-net in base 8, using
- base change [i] based on digital (15, 21, 65540)-net over F16, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- linear OA(1621, 65536, F16, 6) (dual of [65536, 65515, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(1617, 65536, F16, 5) (dual of [65536, 65519, 6]-code), using an extension Ce(4) of the primitive narrow-sense BCH-code C(I) with length 65535 = 164−1, defining interval I = [1,4], and designed minimum distance d ≥ |I|+1 = 5 [i]
- linear OA(160, 4, F16, 0) (dual of [4, 4, 1]-code), using
- discarding factors / shortening the dual code based on linear OA(160, s, F16, 0) (dual of [s, s, 1]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(4) [i] based on
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(1621, 65540, F16, 6) (dual of [65540, 65519, 7]-code), using
(22, 28, large)-Net in Base 8 — Upper bound on s
There is no (22, 28, large)-net in base 8, because
- 4 times m-reduction [i] would yield (22, 24, large)-net in base 8, but