Best Known (30, 41, s)-Nets in Base 8
(30, 41, 822)-Net over F8 — Constructive and digital
Digital (30, 41, 822)-net over F8, using
- net defined by OOA [i] based on linear OOA(841, 822, F8, 11, 11) (dual of [(822, 11), 9001, 12]-NRT-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(841, 4111, F8, 11) (dual of [4111, 4070, 12]-code), using
- discarding factors / shortening the dual code based on linear OA(841, 4112, F8, 11) (dual of [4112, 4071, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(84, 16, F8, 3) (dual of [16, 12, 4]-code or 16-cap in PG(3,8)), using
- construction X applied to Ce(10) ⊂ Ce(6) [i] based on
- discarding factors / shortening the dual code based on linear OA(841, 4112, F8, 11) (dual of [4112, 4071, 12]-code), using
- OOA 5-folding and stacking with additional row [i] based on linear OA(841, 4111, F8, 11) (dual of [4111, 4070, 12]-code), using
(30, 41, 4260)-Net over F8 — Digital
Digital (30, 41, 4260)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(841, 4260, F8, 11) (dual of [4260, 4219, 12]-code), using
- 156 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 34 times 0, 1, 113 times 0) [i] based on linear OA(837, 4100, F8, 11) (dual of [4100, 4063, 12]-code), using
- construction X applied to Ce(10) ⊂ Ce(9) [i] based on
- linear OA(837, 4096, F8, 11) (dual of [4096, 4059, 12]-code), using an extension Ce(10) of the primitive narrow-sense BCH-code C(I) with length 4095 = 84−1, defining interval I = [1,10], and designed minimum distance d ≥ |I|+1 = 11 [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(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(10) ⊂ Ce(9) [i] based on
- 156 step Varšamov–Edel lengthening with (ri) = (2, 6 times 0, 1, 34 times 0, 1, 113 times 0) [i] based on linear OA(837, 4100, F8, 11) (dual of [4100, 4063, 12]-code), using
(30, 41, 6243928)-Net in Base 8 — Upper bound on s
There is no (30, 41, 6243929)-net in base 8, because
- 1 times m-reduction [i] would yield (30, 40, 6243929)-net in base 8, but
- the generalized Rao bound for nets shows that 8m ≥ 1 329228 603273 869978 820015 509225 430728 > 840 [i]