Best Known (31, 31+12, s)-Nets in Base 8
(31, 31+12, 684)-Net over F8 — Constructive and digital
Digital (31, 43, 684)-net over F8, using
- 81 times duplication [i] based on digital (30, 42, 684)-net over F8, using
- net defined by OOA [i] based on linear OOA(842, 684, F8, 12, 12) (dual of [(684, 12), 8166, 13]-NRT-code), using
- OA 6-folding and stacking [i] based on linear OA(842, 4104, F8, 12) (dual of [4104, 4062, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
- construction X applied to Ce(11) ⊂ Ce(9) [i] based on
- 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(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(81, 9, F8, 1) (dual of [9, 8, 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(11) ⊂ Ce(9) [i] based on
- discarding factors / shortening the dual code based on linear OA(842, 4105, F8, 12) (dual of [4105, 4063, 13]-code), using
- OA 6-folding and stacking [i] based on linear OA(842, 4104, F8, 12) (dual of [4104, 4062, 13]-code), using
- net defined by OOA [i] based on linear OOA(842, 684, F8, 12, 12) (dual of [(684, 12), 8166, 13]-NRT-code), using
(31, 31+12, 4011)-Net over F8 — Digital
Digital (31, 43, 4011)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(843, 4011, F8, 12) (dual of [4011, 3968, 13]-code), using
- discarding factors / shortening the dual code based on linear OA(843, 4107, F8, 12) (dual of [4107, 4064, 13]-code), using
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- 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(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(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(81, 10, F8, 1) (dual of [10, 9, 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
- linear OA(80, 1, F8, 0) (dual of [1, 1, 1]-code), using
- dual of repetition code with length 1 [i]
- construction XX applied to Ce(11) ⊂ Ce(9) ⊂ Ce(8) [i] based on
- discarding factors / shortening the dual code based on linear OA(843, 4107, F8, 12) (dual of [4107, 4064, 13]-code), using
(31, 31+12, 1268433)-Net in Base 8 — Upper bound on s
There is no (31, 43, 1268434)-net in base 8, because
- the generalized Rao bound for nets shows that 8m ≥ 680 565435 109296 055602 926282 893398 927160 > 843 [i]