Best Known (138, 138+27, s)-Nets in Base 8
(138, 138+27, 161320)-Net over F8 — Constructive and digital
Digital (138, 165, 161320)-net over F8, using
- 82 times duplication [i] based on digital (136, 163, 161320)-net over F8, using
- net defined by OOA [i] based on linear OOA(8163, 161320, F8, 27, 27) (dual of [(161320, 27), 4355477, 28]-NRT-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8163, 2097161, F8, 27) (dual of [2097161, 2096998, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8163, 2097167, F8, 27) (dual of [2097167, 2097004, 28]-code), using
- construction X applied to Ce(26) ⊂ Ce(24) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(81, 15, F8, 1) (dual of [15, 14, 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(26) ⊂ Ce(24) [i] based on
- discarding factors / shortening the dual code based on linear OA(8163, 2097167, F8, 27) (dual of [2097167, 2097004, 28]-code), using
- OOA 13-folding and stacking with additional row [i] based on linear OA(8163, 2097161, F8, 27) (dual of [2097161, 2096998, 28]-code), using
- net defined by OOA [i] based on linear OOA(8163, 161320, F8, 27, 27) (dual of [(161320, 27), 4355477, 28]-NRT-code), using
(138, 138+27, 1221217)-Net over F8 — Digital
Digital (138, 165, 1221217)-net over F8, using
- embedding of OOA with Gilbert–Varšamov bound [i] based on linear OA(8165, 1221217, F8, 27) (dual of [1221217, 1221052, 28]-code), using
- discarding factors / shortening the dual code based on linear OA(8165, 2097170, F8, 27) (dual of [2097170, 2097005, 28]-code), using
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- linear OA(8162, 2097152, F8, 27) (dual of [2097152, 2096990, 28]-code), using an extension Ce(26) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,26], and designed minimum distance d ≥ |I|+1 = 27 [i]
- linear OA(8148, 2097152, F8, 25) (dual of [2097152, 2097004, 26]-code), using an extension Ce(24) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,24], and designed minimum distance d ≥ |I|+1 = 25 [i]
- linear OA(8141, 2097152, F8, 23) (dual of [2097152, 2097011, 24]-code), using an extension Ce(22) of the primitive narrow-sense BCH-code C(I) with length 2097151 = 87−1, defining interval I = [1,22], and designed minimum distance d ≥ |I|+1 = 23 [i]
- linear OA(81, 16, F8, 1) (dual of [16, 15, 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(81, 2, F8, 1) (dual of [2, 1, 2]-code), using
- dual of repetition code with length 2 [i]
- construction XX applied to Ce(26) ⊂ Ce(24) ⊂ Ce(22) [i] based on
- discarding factors / shortening the dual code based on linear OA(8165, 2097170, F8, 27) (dual of [2097170, 2097005, 28]-code), using
(138, 138+27, large)-Net in Base 8 — Upper bound on s
There is no (138, 165, large)-net in base 8, because
- 25 times m-reduction [i] would yield (138, 140, large)-net in base 8, but