Best Known (35, 35+13, s)-Nets in Base 8
(35, 35+13, 685)-Net over F8 — Constructive and digital
Digital (35, 48, 685)-net over F8, using
- net defined by OOA [i] based on linear OOA(848, 685, F8, 13, 13) (dual of [(685, 13), 8857, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(848, 4111, F8, 13) (dual of [4111, 4063, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(833, 4096, F8, 10) (dual of [4096, 4063, 11]-code), using an extension Ce(9) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,9], and designed minimum distance d ≥ |I|+1 = 10 [i]
- linear OA(83, 15, F8, 2) (dual of [15, 12, 3]-code), using
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 63 = 82−1, defining interval I = [0,1], and designed minimum distance d ≥ |I|+1 = 3 [i]
- discarding factors / shortening the dual code based on linear OA(83, 63, F8, 2) (dual of [63, 60, 3]-code), using
- construction X applied to Ce(12) ⊂ Ce(9) [i] based on
- OOA 6-folding and stacking with additional row [i] based on linear OA(848, 4111, F8, 13) (dual of [4111, 4063, 14]-code), using
(35, 35+13, 4152)-Net over F8 — Digital
Digital (35, 48, 4152)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(848, 4152, F8, 13) (dual of [4152, 4104, 14]-code), using
- 49 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 41 times 0) [i] based on linear OA(845, 4100, F8, 13) (dual of [4100, 4055, 14]-code), using
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- linear OA(845, 4096, F8, 13) (dual of [4096, 4051, 14]-code), using an extension Ce(12) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,12], and designed minimum distance d ≥ |I|+1 = 13 [i]
- linear OA(841, 4096, F8, 12) (dual of [4096, 4055, 13]-code), using an extension Ce(11) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,11], and designed minimum distance d ≥ |I|+1 = 12 [i]
- linear OA(80, 4, F8, 0) (dual of [4, 4, 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
- construction X applied to Ce(12) ⊂ Ce(11) [i] based on
- 49 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 41 times 0) [i] based on linear OA(845, 4100, F8, 13) (dual of [4100, 4055, 14]-code), using
(35, 35+13, 5073745)-Net in Base 8 — Upper bound on s
There is no (35, 48, 5073746)-net in base 8, because
- 1 times m-reduction [i] would yield (35, 47, 5073746)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 2 787595 786782 025836 959695 727655 938557 432568 > 847 [i]