Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes
Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><...
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| Format: | Article |
| Language: | English |
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MDPI AG
2024-01-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/12/3/443 |
| _version_ | 1797318458903363584 |
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| author | Aleksandar Minja Vojin Šenk |
| author_facet | Aleksandar Minja Vojin Šenk |
| author_sort | Aleksandar Minja |
| collection | DOAJ |
| description | Many families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></semantics></math></inline-formula> ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. Generalized Kerdock codes represent an extension of classical Kerdock codes to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> ring. In this paper, we develop two novel soft-input decoders, designed to exploit the unique structure of these codes. We introduce a novel soft-input ML decoding algorithm and a soft-input soft-output MAP decoding algorithm of generalized Kerdock codes, with a complexity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mi>S</mi></msup><msub><mo form="prefix">log</mo><mn>2</mn></msub><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>N</i> is the length of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> code, that is, the number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> symbols in a codeword. Simulations show that our novel decoders outperform the classical lifting decoder in terms of error rate by some 5 dB. |
| first_indexed | 2024-03-08T03:52:41Z |
| format | Article |
| id | doaj.art-6019a99434a149b6b5597a4bf8f15ce3 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-08T03:52:41Z |
| publishDate | 2024-01-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-6019a99434a149b6b5597a4bf8f15ce32024-02-09T15:18:22ZengMDPI AGMathematics2227-73902024-01-0112344310.3390/math12030443Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock CodesAleksandar Minja0Vojin Šenk1Department of Power, Electronic and Telecommunication Engineering, Faculty of Engineering (Technical Sciences), University of Novi Sad, 21000 Novi Sad, SerbiaDepartment of Power, Electronic and Telecommunication Engineering, Faculty of Engineering (Technical Sciences), University of Novi Sad, 21000 Novi Sad, SerbiaMany families of binary nonlinear codes (e.g., Kerdock, Goethals, Delsarte–Goethals, Preparata) can be very simply constructed from linear codes over the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>4</mn></msub></semantics></math></inline-formula> ring (ring of integers modulo 4), by applying the Gray map to the quaternary symbols. Generalized Kerdock codes represent an extension of classical Kerdock codes to the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> ring. In this paper, we develop two novel soft-input decoders, designed to exploit the unique structure of these codes. We introduce a novel soft-input ML decoding algorithm and a soft-input soft-output MAP decoding algorithm of generalized Kerdock codes, with a complexity of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi mathvariant="script">O</mi><mo>(</mo><msup><mi>N</mi><mi>S</mi></msup><msub><mo form="prefix">log</mo><mn>2</mn></msub><mi>N</mi><mo>)</mo></mrow></semantics></math></inline-formula>, where <i>N</i> is the length of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> code, that is, the number of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> symbols in a codeword. Simulations show that our novel decoders outperform the classical lifting decoder in terms of error rate by some 5 dB.https://www.mdpi.com/2227-7390/12/3/443codes over ringsKerdock codesMAP decodingMonte Carlo simulation |
| spellingShingle | Aleksandar Minja Vojin Šenk Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes Mathematics codes over rings Kerdock codes MAP decoding Monte Carlo simulation |
| title | Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes |
| title_full | Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes |
| title_fullStr | Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes |
| title_full_unstemmed | Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes |
| title_short | Decoding of <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><msup><mn>2</mn><mi>S</mi></msup></msub></semantics></math></inline-formula> Linear Generalized Kerdock Codes |
| title_sort | decoding of inline formula math display inline semantics msub mi mathvariant double struck z mi msup mn 2 mn mi s mi msup msub semantics math inline formula linear generalized kerdock codes |
| topic | codes over rings Kerdock codes MAP decoding Monte Carlo simulation |
| url | https://www.mdpi.com/2227-7390/12/3/443 |
| work_keys_str_mv | AT aleksandarminja decodingofinlineformulamathdisplayinlinesemanticsmsubmimathvariantdoublestruckzmimsupmn2mnmismimsupmsubsemanticsmathinlineformulalineargeneralizedkerdockcodes AT vojinsenk decodingofinlineformulamathdisplayinlinesemanticsmsubmimathvariantdoublestruckzmimsupmn2mnmismimsupmsubsemanticsmathinlineformulalineargeneralizedkerdockcodes |