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"maxValue": "simplex_minimum_distance", "unitTests": [], "correctAnswerStyle": "plain", "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "variableReplacements": [], "marks": "0.5", "mustBeReducedPC": 0}], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "variableReplacements": [], "marks": 0, "sortAnswers": false}], "statement": "<p>Write down the word-length, dimension and minimum distance of $C$, where</p>", "tags": [], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "extensions": ["codewords", "permutations"], "variablesTest": {"condition": "", "maxRuns": 100}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Compute the word length, minimum distance and dimension of some given Hamming codes.</p>"}, "advice": "<h4>a)</h4>\n<p>$\\operatorname{Ham}(\\var{binary_hamming_rows})$ has word-length $\\simplify[]{2^{binary_hamming_rows}-1 = {binary_word_length}}$, dimension $\\simplify[]{{binary_word_length}-{binary_hamming_rows} = {binary_dimension}}$ and minimum distance $3$.</p>\n<h4>b)</h4>\n<p>$\\operatorname{Ham}_{\\var{hamming_prime}}(\\var{hamming_rows})$ has word-length $\\simplify[]{({hamming_prime}^{hamming_rows}-1)/({hamming_prime}-1) = {hamming_word_length}}$, dimension $\\simplify[]{{hamming_word_length}-{hamming_rows} = {hamming_dimension}}$ and minimum distance $3$.</p>\n<h4>c)</h4>\n<p>$\\operatorname{Ham}^+(\\var{parity_hamming_rows})$ has word-length $\\simplify[]{2^{parity_hamming_rows} = {parity_hamming_word_length}}$, dimension $\\simplify[]{{parity_hamming_word_length}-1-{parity_hamming_rows} = {parity_hamming_dimension}}$ and minimum distance $4$.</p>\n<h4>d)</h4>\n<p>$\\operatorname{Sim}_{\\var{simplex_prime}}(\\var{simplex_rows})$ has word-length $\\simplify[]{({simplex_prime}^{simplex_rows}-1)/({simplex_prime}-1) = {simplex_word_length}}$, dimension $\\var{simplex_rows}$ and minimum distance $\\simplify[]{{simplex_prime}^({simplex_rows}-1)={simplex_minimum_distance}}$.</p>", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas-editor.mas.ncl.ac.uk/accounts/profile/3/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas-editor.mas.ncl.ac.uk/accounts/profile/3/"}, {"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}]}