Integer Codes Correcting Asymmetric Errors in Nand Flash Memory

• Main Authors: Hristo Kostadinov, Nikolai Manev
• Format: Article
• Language: English
• Published: MDPI AG 2021-06-01
• Series: Mathematics
• Subjects: integer codes; flash memory; asymmetric errors
• Online Access: https://www.mdpi.com/2227-7390/9/11/1269

Memory devices based on floating-gate transistor have recently become dominant technology for non-volatile storage devices like USB flash drives, memory cards, solid-state disks, etc. In contrast to many communication channels, the errors observed in flash memory device use are not random but of special, mainly asymmetric, type. Integer codes which have proved their efficiency in many cases with asymmetric errors can be applied successfully to flash memory devices, too. This paper presents a new construction and integer codes over a ring of integers modulo <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>A</mi><mo>=</mo><msup><mn>2</mn><mi>n</mi></msup><mo>+</mo><mn>1</mn></mrow></semantics></math></inline-formula> capable of correcting single errors of type <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo><mo>,</mo><mspace width="4pt"></mspace><mo>(</mo><mo>±</mo><mn>1</mn><mo>,</mo><mo>±</mo><mn>2</mn><mo>)</mo></mrow></semantics></math></inline-formula>, or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></semantics></math></inline-formula> that are typical for flash memory devices. The construction is based on the use of cyclotomic cosets of 2 modulo <i>A</i>. The parity-check matrices of the codes are listed for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≤</mo><mn>10</mn><mo>.</mo></mrow></semantics></math></inline-formula>