Binary Fuzzy Codes and Some Properties of Hamming Distance of Fuzzy Codes

Abstract

In this paper, we discussed binary fuzzy codes over a vector space <inline-formula id="M1"> <math id="mathml_M1" display="inline" overflow="scroll"><msubsup><mi>F</mi><mn>2</mn><mi>n</mi></msubsup><mspace width="thickmathspace"></mspace></math></inline-formula> by relating classical codes with the probability of a binary symmetric channel (BSC) for receiving a sent codeword correctly. We used the weight of error patterns between a received word and the possible sent codewords to define fuzzy words over <inline-formula id="M2"> <math id="mathml_M2" display="inline" overflow="scroll"><mi>n</mi></math></inline-formula>-dimensional vector space <inline-formula id="M3"> <math id="mathml_M3" display="inline" overflow="scroll"><msubsup><mi>F</mi><mn>2</mn><mi>n</mi></msubsup></math></inline-formula>, and used it to define binary fuzzy codes. We also added some properties of Hamming distance of binary fuzzy codes, and the bounds of a Hamming distance of binary fuzzy codes for <inline-formula id="M4"> <math id="mathml_M4" display="inline" overflow="scroll"><mi>p</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mrow class="MJX-TeXAtom-ORD"><mi>r</mi></mrow></math></inline-formula>, where <inline-formula id="M5"> <math id="mathml_M5" display="inline" overflow="scroll"><mi>r</mi><mo>⩾</mo><mn>3</mn></math></inline-formula>, and <inline-formula id="M6"> <math id="mathml_M6" display="inline" overflow="scroll"><mi>r</mi><mo>∈</mo><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">Z</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mo>+</mo></mrow></msup></math></inline-formula>, are determined. Finding Hamming distance of binary fuzzy codes is used for decoding sent messages on a BSC.