// Numbas version: exam_results_page_options
{"name": "Hamming distances", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Hamming distances", "tags": [], "metadata": {"description": "<p>Short question to test encoding some random binary strings using repetition and parity bit. Randomised 5-bit strings.</p>", "licence": "None specified"}, "statement": "<p>A set of words in $\\mathbb{Z}_2^5$ is described below. You will be asked to encode them, calculate the minimum distance between code words, and describe how many errors can be detected or corrected.</p>\n<p>Encoding is described as follows:</p>\n<ul>\n<li>First, repeat the word. So if $w = 01101$, then we would get $0110101101$.</li>\n<li>Add a parity bit to the end based on the original word. Since $\\mathrm{wt}(w) = 3$, then the parity bit is $1$.</li>\n<li>So $E(w) = 01101011011$.</li>\n</ul>\n<p>The set of words is $W = \\{ \\var{binstring1}, \\var{binstring2}, \\var{binstring3}, \\var{binstring4}\\}$.</p>", "advice": "<p>The minimum distance is the smallest non-zero value of $d(x,y)$ where $x$, $y \\in C$. Call this $m$.</p>\n<p>If the minimum distance of the code is $k$, then the code can detect up to $m-1$ errors.</p>\n<p>If the minimum distance of the code is $k$, then the code can correct up to $\\lfloor \\frac{m-1}{2} \\rfloor$.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"binstring1": {"name": "binstring1", "group": "Ungrouped variables", "definition": "tobinary(random(16..19))", "description": "", "templateType": "anything", "can_override": false}, "binstring2": {"name": "binstring2", "group": "Ungrouped variables", "definition": "tobinary(random(20..23))", "description": "", "templateType": "anything", "can_override": false}, "binstring3": {"name": "binstring3", "group": "Ungrouped variables", "definition": "tobinary(random(24..27))", "description": "", "templateType": "anything", "can_override": false}, "binstring4": {"name": "binstring4", "group": "Ungrouped variables", "definition": "tobinary(random(28..31))", "description": "", "templateType": "anything", "can_override": false}, "bins1": {"name": "bins1", "group": "Ungrouped variables", "definition": "map(dec(binstring1[k]),k,0..4)", "description": "", "templateType": "anything", "can_override": false}, "bins2": {"name": "bins2", "group": "Ungrouped variables", "definition": "map(dec(binstring2[k]),k,0..4)", "description": "", "templateType": "anything", "can_override": false}, "bins3": {"name": "bins3", "group": "Ungrouped variables", "definition": "map(dec(binstring3[k]),k,0..4)", "description": "", "templateType": "anything", "can_override": false}, "bins4": {"name": "bins4", "group": "Ungrouped variables", "definition": "map(dec(binstring4[k]),k,0..4)", "description": "", "templateType": "anything", "can_override": false}, "err12": {"name": "err12", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code1[k])+dec(code2[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "err13": {"name": "err13", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code1[k])+dec(code3[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "err14": {"name": "err14", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code1[k])+dec(code4[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "err23": {"name": "err23", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code2[k])+dec(code3[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "err24": {"name": "err24", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code2[k])+dec(code4[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "err34": {"name": "err34", "group": "Ungrouped variables", "definition": "sum(map(mod(dec(code3[k])+dec(code4[k]),2),k,0..10))", "description": "", "templateType": "anything", "can_override": false}, "minerror": {"name": "minerror", "group": "Ungrouped variables", "definition": "min(errors)", "description": "", "templateType": "anything", "can_override": false}, "errors": {"name": "errors", "group": "Ungrouped variables", "definition": "[err12,err13,err14,err23,err24,err34]", "description": "", "templateType": "anything", "can_override": false}, "detect": {"name": "detect", "group": "Ungrouped variables", "definition": "if(minerror-1 < 0, 0, minerror -1)", "description": "", "templateType": "anything", "can_override": false}, "correct": {"name": "correct", "group": "Ungrouped variables", "definition": "if((minerror-1) < 0, 0, floor((minerror-1)/2))", "description": "", "templateType": "anything", "can_override": false}, "code1": {"name": "code1", "group": "Ungrouped variables", "definition": "bins1 + bins1 + par12", "description": "", "templateType": "anything", "can_override": false}, "code2": {"name": "code2", "group": "Ungrouped variables", "definition": "bins2 + bins2 + par22", "description": "", "templateType": "anything", "can_override": false}, "code3": {"name": "code3", "group": "Ungrouped variables", "definition": "bins3 + bins3 + par32", "description": "", "templateType": "anything", "can_override": false}, "code4": {"name": "code4", "group": "Ungrouped variables", "definition": "bins4 + bins4 + par42", "description": "", "templateType": "anything", "can_override": false}, "par12": {"name": "par12", "group": "Ungrouped variables", "definition": "mod(sum(bins1),2)", "description": "", "templateType": "anything", "can_override": false}, "par22": {"name": "par22", "group": "Ungrouped variables", "definition": "mod(sum(bins2),2)", "description": "", "templateType": "anything", "can_override": false}, "par32": {"name": "par32", "group": "Ungrouped variables", "definition": "mod(sum(bins3),2)", "description": "", "templateType": "anything", "can_override": false}, "par42": {"name": "par42", "group": "Ungrouped variables", "definition": "mod(sum(bins4),2)", "description": "", "templateType": "anything", "can_override": false}, "codes1": {"name": "codes1", "group": "Ungrouped variables", "definition": "map(string(code1[k]),k,0..10)", "description": "", "templateType": "anything", "can_override": false}, "codeword1": {"name": "codeword1", "group": "Ungrouped variables", "definition": "codes1[0] + codes1[1] + codes1[2] + codes1[3] + codes1[4] + codes1[5] + codes1[6] + codes1[7] + codes1[8] + codes1[9] + codes1[10]", "description": "", "templateType": "anything", "can_override": false}, "codeword2": {"name": "codeword2", "group": "Ungrouped variables", "definition": "codes2[0] + codes2[1] + codes2[2] + codes2[3] + codes2[4] + codes2[5] + codes2[6] + codes2[7] + codes2[8] + codes2[9] + codes2[10]", "description": "", "templateType": "anything", "can_override": false}, "codes2": {"name": "codes2", "group": "Ungrouped variables", "definition": "map(string(code2[k]),k,0..10)", "description": "", "templateType": "anything", "can_override": false}, "codes3": {"name": "codes3", "group": "Ungrouped variables", "definition": "map(string(code3[k]),k,0..10)", "description": "", "templateType": "anything", "can_override": false}, "codes4": {"name": "codes4", "group": "Ungrouped variables", "definition": "map(string(code4[k]),k,0..10)", "description": "", "templateType": "anything", "can_override": false}, "codeword3": {"name": "codeword3", "group": "Ungrouped variables", "definition": "codes3[0] + codes3[1] + codes3[2] + codes3[3] + codes3[4] + codes3[5] + codes3[6] + codes3[7] + codes3[8] + codes3[9] + codes3[10]", "description": "", "templateType": "anything", "can_override": false}, "codeword4": {"name": "codeword4", "group": "Ungrouped variables", "definition": "codes4[0] + codes4[1] + codes4[2] + codes4[3] + codes4[4] + codes4[5] + codes4[6] + codes4[7] + codes4[8] + codes4[9] + codes4[10]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["binstring1", "binstring2", "binstring3", "binstring4", "bins1", "bins2", "bins3", "bins4", "err12", "err13", "err14", "err23", "err24", "err34", "minerror", "errors", "detect", "correct", "code1", "code2", "code3", "code4", "par12", "par22", "par32", "par42", "codes1", "codes2", "codes3", "codes4", "codeword1", "codeword2", "codeword3", "codeword4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Encode $\\var{binstring1}$.</p>", "answer": "{codeword1}", "displayAnswer": "", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Encode $\\var{binstring2}$.</p>", "answer": "{codeword2}", "displayAnswer": "", "matchMode": "exact"}, {"type": "patternmatch", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Encode $\\var{binstring3}$.</p>", "answer": 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(The code words are the $11$-bit binary strings you have just found.)</p>", "minValue": "{minerror}", "maxValue": "{minerror}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>How many errors can this code detect?</p>", "minValue": "{detect}", "maxValue": "{detect}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>How many errors can this code correct?</p>", "minValue": "{correct}", "maxValue": "{correct}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}]}]}], "contributors": [{"name": "Alexander Corner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/5328/"}]}