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2026
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| Online Access: | https://doi.org/10.5281/zenodo.19508053 |
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| author | Kulkarni, Raghu |
| author_facet | Kulkarni, Raghu |
| contents | <p><span><span class="citation-208">An old question in sphere-packing turns out to yield an unexpected construction in quantum error correction</span></span><span><span class="citation-208 citation-end-208"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-207">The face-centered cubic lattice densest sphere packing in three dimensions, proved by Hales generates a CSS code with encoding rate 2/3</span></span><span><span class="citation-207 citation-end-207"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-206">We show that this construction extends to every Dn root lattice (n >= 3), yielding codes with parameters [n(n-1)L^n/2, (n-2)(n+1)L^n/2 + 2, 3] and encoding rate converging to 1 - 4/K as L -> infinity where K = 2n(n-1) is the coordination number</span></span><span><span class="citation-206 citation-end-206"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-205">CSS validity, the rank formula, and the rate are proved analytically for all n >= 3</span></span><span><span class="citation-205 citation-end-205"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-204">The first two members are computationally verified: the D3 (FCC) code and the D4 code [1536, 1282, 3] at L=4 confirmed at L=6 as [7776, 6482, 3]</span></span><span><span class="citation-204 citation-end-204"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-203">For D4 at L=4, all 4096 triangles are logical operators, establishing d=3 exactly</span></span><span><span class="citation-203 citation-end-203"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-202">As K grows along the densest-packing hierarchy the protected fraction approaches unity, making this, to our knowledge, the first proven infinite family of topological CSS codes whose rate converges to 1 at fixed distance</span></span><span><span class="citation-202 citation-end-202"><sup class="superscript"></sup></span></span><span>.</span></p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19508053 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | CSS Codes from Dn Root Lattices: An Infinite Family with Rate 1 - 4/K Kulkarni, Raghu <p><span><span class="citation-208">An old question in sphere-packing turns out to yield an unexpected construction in quantum error correction</span></span><span><span class="citation-208 citation-end-208"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-207">The face-centered cubic lattice densest sphere packing in three dimensions, proved by Hales generates a CSS code with encoding rate 2/3</span></span><span><span class="citation-207 citation-end-207"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-206">We show that this construction extends to every Dn root lattice (n >= 3), yielding codes with parameters [n(n-1)L^n/2, (n-2)(n+1)L^n/2 + 2, 3] and encoding rate converging to 1 - 4/K as L -> infinity where K = 2n(n-1) is the coordination number</span></span><span><span class="citation-206 citation-end-206"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-205">CSS validity, the rank formula, and the rate are proved analytically for all n >= 3</span></span><span><span class="citation-205 citation-end-205"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-204">The first two members are computationally verified: the D3 (FCC) code and the D4 code [1536, 1282, 3] at L=4 confirmed at L=6 as [7776, 6482, 3]</span></span><span><span class="citation-204 citation-end-204"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-203">For D4 at L=4, all 4096 triangles are logical operators, establishing d=3 exactly</span></span><span><span class="citation-203 citation-end-203"><sup class="superscript"></sup></span></span><span>. </span><span><span class="citation-202">As K grows along the densest-packing hierarchy the protected fraction approaches unity, making this, to our knowledge, the first proven infinite family of topological CSS codes whose rate converges to 1 at fixed distance</span></span><span><span class="citation-202 citation-end-202"><sup class="superscript"></sup></span></span><span>.</span></p> |
| title | CSS Codes from Dn Root Lattices: An Infinite Family with Rate 1 - 4/K |
| url | https://doi.org/10.5281/zenodo.19508053 |