Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes
Lattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their conste...
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MDPI AG
2025-05-01
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| author | Perathorn Pooksombat Wittawat Kositwattanarerk |
| author_facet | Perathorn Pooksombat Wittawat Kositwattanarerk |
| author_sort | Perathorn Pooksombat |
| collection | DOAJ |
| description | Lattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their constellation usually involves superfluous elements that need to be mapped to a representative within the same coset to reduce average transmission power. One such elegant shaping function is a componentwise modulo, which is known to produce a cubic shaping for Barnes–Wall lattices. In this paper, we generalize this result to lattices over quadratic rings of integers, thus encompassing constructions from <i>p</i>-ary codes, where <i>p</i> is a prime number. We identify all bases that permit cubic modulo shaping. This provides useful insights into practical encoding and decoding of lattice codes from multi-level constructions. |
| format | Article |
| id | doaj-art-7206365aeebc416b94d237df54674cfc |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-05-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-7206365aeebc416b94d237df54674cfc2025-08-20T02:33:47ZengMDPI AGMathematics2227-73902025-05-011310156210.3390/math13101562Cubic Shaping of Lattice Constellations from Multi-Level Constructions from CodesPerathorn Pooksombat0Wittawat Kositwattanarerk1The Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, ThailandThe Department of Mathematics, Faculty of Science, Mahidol University, Bangkok 10400, ThailandLattice codes play an important role in wireless communication and are closely related to linear codes. Multi-level constructions of complex lattices from codes are known to produce lattice codes with desirable parameters and efficient encoding and decoding of information bits. However, their constellation usually involves superfluous elements that need to be mapped to a representative within the same coset to reduce average transmission power. One such elegant shaping function is a componentwise modulo, which is known to produce a cubic shaping for Barnes–Wall lattices. In this paper, we generalize this result to lattices over quadratic rings of integers, thus encompassing constructions from <i>p</i>-ary codes, where <i>p</i> is a prime number. We identify all bases that permit cubic modulo shaping. This provides useful insights into practical encoding and decoding of lattice codes from multi-level constructions.https://www.mdpi.com/2227-7390/13/10/1562latticeslattice codeslattice constellationcubic shapingtiling |
| spellingShingle | Perathorn Pooksombat Wittawat Kositwattanarerk Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes Mathematics lattices lattice codes lattice constellation cubic shaping tiling |
| title | Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes |
| title_full | Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes |
| title_fullStr | Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes |
| title_full_unstemmed | Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes |
| title_short | Cubic Shaping of Lattice Constellations from Multi-Level Constructions from Codes |
| title_sort | cubic shaping of lattice constellations from multi level constructions from codes |
| topic | lattices lattice codes lattice constellation cubic shaping tiling |
| url | https://www.mdpi.com/2227-7390/13/10/1562 |
| work_keys_str_mv | AT perathornpooksombat cubicshapingoflatticeconstellationsfrommultilevelconstructionsfromcodes AT wittawatkositwattanarerk cubicshapingoflatticeconstellationsfrommultilevelconstructionsfromcodes |