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2025年1月27日 (月) 02:26時点における版

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問4(問題文)

　符号長7ビット、情報ビット数4ビットのハミング符号による誤り訂正の方法を、次のとおりとする。

受信した7ビットの符号語x₁x₂x₃x₄x₅x₆x₇(x_k=0又は1)に対して

c₀ ＝ x₁ ＋x₃ ＋x₅ ＋x₇

c₁ ＝ x₂＋x₃ ＋x₆＋x₇

c₂ ＝ x₄＋x₅＋x₆＋x₇

(いずれも mod 2 での計算)

を計算し、c₀、c₁、c₂の中に少なくとも一つは0でないものがある場合には、

i = c₀＋c₁×2＋c₂×4

を求めて、左から i ビット目を反転することによって誤りを訂正する。

受信した符号語が1000101であった場合、誤り訂正後の符号語はどれか。

ア　1000001

イ　1000101

ウ　1001101

エ　1010101

回答・解説

　誤り符号の仕組みを知ってるかを問いたいのでしょうけれど、書かれたとおり計算すると答えれる四則演算できたもん勝ちの問題です。やってみましょう。といってもmod 2が2で割った余りっていう意味を理解できないと駄目ですけどね。

c₀ ＝ x₁ ＋x₃ ＋x₅ ＋x₇は

なので、

1+0+1+1=3、3 ÷ 2=1余り1、余りが格納されるので、c0=1

c₁ ＝ x₂＋x₃ ＋x₆＋x₇

なので、

0+0+0+1=1、1 ÷ 2=0余り1、つまりc1=1

c₂ ＝ x₄＋x₅＋x₆＋x₇

なので

0+1+0+1=2、1 ÷ 2=2余り1、つまりc2=0

c0、c1、c2に0でないものが含まれるので、引き続き誤り訂正処理をします。

i = c₀＋c₁×2＋c₂×4

なので、

1+1×2+0+0×4=3、i=3

1000101の左から 3 ビット目を反転を反転して、1010101

したがって

エ　1010101

　が答えです。

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-\( 1+0+1+1 = 3 \)、\( 3 \divisionsymbol 2 = 1 \text{余り} 1 \)、余りが格納されるので、\( c_0 = 1 \)
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@@ 93行目: / 107行目: @@
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@@ 102行目: / 116行目: @@
-\( 0+1+0+1 = 2 \)、\( 1 \divisionsymbol 2 = 2 \text{余り} 1 \)、つまり\( c_2 = 0 \)
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_0+1+0+1_=_2" style="width: 9.257em; display: inline-block;"><span style="display: inline-block; position: relative; width: 7.263em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1007.21em, 2.355em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">0</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">0</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">2</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.127em; border-left: 0px solid; width: 0px; height: 1.107em;"></span></span></span>、<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_1_\divisionsymbol_2_=_2_\text{余り}_1" style="width: 8.541em; display: inline-block;"><span style="display: inline-block; position: relative; width: 6.701em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1006.5em, 2.61em, -999.997em); top: -2.247em; left: 0em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">1</span><span class="texatom"><span class="mrow"><span class="mtext" style="font-family: NeoEulerMathJax_Main;">&nbsp;</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">÷</span><span class="mtext" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">&nbsp;</span></span></span><span class="mn" style="font-family: NeoEulerMathJax_Main;">2</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">2</span><span class="mtext"><span style="font-family: Meiryo, STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 79%; font-style: normal; font-weight: normal;">余</span><span style="font-family: Meiryo, STIXGeneral, &quot;Arial Unicode MS&quot;, serif; font-size: 79%; font-style: normal; font-weight: normal;">り</span></span><span class="mn" style="font-family: NeoEulerMathJax_Main;">1</span></span><span style="display: inline-block; width: 0px; height: 2.252em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.321em; border-left: 0px solid; width: 0px; height: 1.432em;"></span></span></span>、つまり<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_c\_2_=_0" style="width: 3.479em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.712em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1002.66em, 2.457em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px;"><span style="position: absolute; clip: rect(3.377em, 1000.46em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">c</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">0</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 1.237em;"></span></span></span>
-\( c_0 \)、\( c_1 \)、\( c_2 \)に0でないものが含まれるので、引き続き誤り訂正処理をします。
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_c\_0" style="width: 1.179em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.485em, 1000.92em, 2.406em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px;"><span style="position: absolute; clip: rect(3.377em, 1000.46em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">c</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">0</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 0.912em;"></span></span></span>、<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_c\_1" style="width: 1.179em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.485em, 1000.92em, 2.406em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px;"><span style="position: absolute; clip: rect(3.377em, 1000.46em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">c</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">1</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 0.912em;"></span></span></span>、<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_c\_2" style="width: 1.179em; display: inline-block;"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.485em, 1000.92em, 2.406em, -999.997em); top: -2.094em; left: 0em;"><span class="mrow"><span class="msubsup"><span style="display: inline-block; position: relative; width: 0.923em; height: 0px;"><span style="position: absolute; clip: rect(3.377em, 1000.46em, 4.144em, -999.997em); top: -3.986em; left: 0em;"><span class="mi" style="font-family: NeoEulerMathJax_Main;">c</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span><span style="position: absolute; top: -3.832em; left: 0.514em;"><span class="mn" style="font-size: 70.7%; font-family: NeoEulerMathJax_Main;">2</span><span style="display: inline-block; width: 0px; height: 3.991em;"></span></span></span></span></span><span style="display: inline-block; width: 0px; height: 2.099em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.256em; border-left: 0px solid; width: 0px; height: 0.912em;"></span></span></span>に0でないものが含まれるので、引き続き誤り訂正処理をします。
@@ 114行目: / 128行目: @@
-\( 1 + 1 \times 2 + 0 + 0 \times 4 = 3 \)、\( i = 3 \)
+<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_1_+_1_\times_2_+_0_+_0_\times_4_=_3" style="width: 13.501em; display: inline-block;"><span style="display: inline-block; position: relative; width: 10.638em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1010.59em, 2.355em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mn" style="font-family: NeoEulerMathJax_Main;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">1</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">×</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">2</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">0</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">+</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">0</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">×</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.207em;">4</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">3</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.127em; border-left: 0px solid; width: 0px; height: 1.107em;"></span></span></span>、<span class="MathJax_Preview" style="color: inherit;"></span><span class="MathJax" tabindex="0" style="position: relative;" role="presentation"><span class="none-math" id="Inline:_i_=_3" style="width: 2.764em; display: inline-block;"><span style="display: inline-block; position: relative; width: 2.15em; height: 0px; font-size: 127%;"><span style="position: absolute; clip: rect(1.281em, 1002.1em, 2.303em, -999.997em); top: -2.145em; left: 0em;"><span class="mrow"><span class="mi" style="font-family: NeoEulerMathJax_Main;">i</span><span class="mo" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">=</span><span class="mn" style="font-family: NeoEulerMathJax_Main; padding-left: 0.258em;">3</span></span><span style="display: inline-block; width: 0px; height: 2.15em;"></span></span></span><span style="display: inline-block; overflow: hidden; vertical-align: -0.062em; border-left: 0px solid; width: 0px; height: 1.042em;"></span></span></span>