MinT - Best Known (137−23, 137, s)-Nets in Base 8

Best Known (137−23, 137, s)-Nets in Base 8

(137−23, 137, 47664)-Net over F₈ — Constructive and digital

Digital (114, 137, 47664)-net over F₈, using

net defined by OOA [i] based on linear OOA(8¹³⁷, 47664, F₈, 23, 23) (dual of [(47664, 23), 1096135, 24]-NRT-code), using
- OOA 11-folding and stacking with additional row [i] based on linear OA(8¹³⁷, 524305, F₈, 23) (dual of [524305, 524168, 24]-code), using
  - 1 times code embedding in larger space [i] based on linear OA(8¹³⁶, 524304, F₈, 23) (dual of [524304, 524168, 24]-code), using
    - trace code [i] based on linear OA(64⁶⁸, 262152, F₆₄, 23) (dual of [262152, 262084, 24]-code), using
      - constructionÂ X applied to C([0,11]) âŠ‚ C([0,10]) [i] based on
        linear OA(64⁶⁷, 262145, F₆₄, 23) (dual of [262145, 262078, 24]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,11], and minimum distance dÂ â‰¥ |{−11,−10,…,11}|+1Â =Â 24 (BCH-bound) [i]
        linear OA(64⁶¹, 262145, F₆₄, 21) (dual of [262145, 262084, 22]-code), using the expurgated narrow-sense BCH-code C(I) with length 262145Â | 64⁶âˆ’1, defining interval IÂ =Â [0,10], and minimum distance dÂ â‰¥ |{−10,−9,…,10}|+1Â =Â 22 (BCH-bound) [i]
        linear OA(64¹, 7, F₆₄, 1) (dual of [7, 6, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(64¹, s, F₆₄, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]

(137−23, 137, 543940)-Net over F₈ — Digital

Digital (114, 137, 543940)-net over F₈, using

Gilbertâ€“VarÅ¡amov bound [i]

(137−23, 137, large)-Net in Base 8 — Upper bound on s

There is no (114, 137, large)-net in base 8, because

21 times m-reduction [i] would yield (114, 116, large)-net in base 8, but
- mutually orthogonal hypercube bound [i]