If \$N\$ is the number of bits then your input signal range is divided into quantization intervals of size
$$q = \frac{V_{ref}}{2^N}$$
The maximum quantization error is \$q/2\$ and it is usually assumed that the quantization error is uniformly distributed between \$-q/2\$ and \$q/2\$. So the PDF of the quantization error is constant between \$-q/2\$ and \$q/2\$ with height \$1/q\$. With this assumption the quantization noise power is
$$P_q = \frac{1}{q}\int_{-q/2}^{q/2}x^2dx = \frac{q^2}{12} = \frac{V_{ref}^2}{12\cdot 2^{2N}}$$
The signal power is given by
$$P_x = \frac{V_{max}^2}{2} \text{ with } V_{max} = 100mV$$
The factor \$1/2\$ in the signal power comes from the fact that the input is sinusoidal, i.e. its power is given by half its maximum value. Note that the maximum value is the amplitude which is half the peak-to-peak value. Putting everything together we get
$$SNR = 10\log\frac{P_x}{P_q} = 10\log\frac{6V_{max}^22^{2N}}{V_{ref}^2} = \\
= 10\log\frac{6V_{max}^2}{V_{ref}^2} + 10\log 2^{2N} =
10\log\frac{6(0.1)^2}{5^2} + N\cdot 20\log 2 =\\
= -26.2 + 12\cdot 6.02 = 46\text{dB}$$
EDIT: To see how the \$1.76\$dB pop up we now use \$V_{pp}\$ instead of \$V_{max}\$, where \$V_{pp}=2V_{max}\$ is the peak-to-peak input voltage. The SNR can then be written as
$$SNR = 10\log\frac{3V_{pp}^2}{2V_{ref}^2} + 10\log 2^{2N} =\\
= 10\log\frac{V_{pp}^2}{V_{ref}^2} + 10\log\frac{3}{2} + N\cdot 20\log 2 =\\
= 10\log\frac{V_{pp}^2}{V_{ref}^2} + 1.76 + 6.02\cdot N$$
So if we use the maximum input range, i.e. \$V_{pp}=V_{ref}\$ we get the formula that was mentioned in the question.
What your professor is almost certainly referring to is SSDS (Spread Spectrum Direct Sequence) encoding technicals (yes they are convolutional) and they trade off data rate for a given bandwidth for the ability to recover signal in deep noise. CDMA systems from Qualcomm use CDMA but not so much for signal in noise recovery but for code diversity in band, meaning multiple independent transmissions overlapped in band.
CDMA uses the concept of process gain to reflect the ability to recover from noise and it counts on the fact that the "pseudo noise" (aka PN) has a distinct matched filter. When the matched filter is applied any noise in band will be mapped out of band and any PN will be mapped to a modulated signal. the process gain (measured in dB) tells you how much excess that you can have in S/N.
However, 60dB is trivial in context to the the NASA PN codes that are used in it's deep space network. These codes can have repeat lengths of 1000's of chips. A chip is s sub-unit of modulation, it is the ratio of chips to bits that determines the process gain.
An excellent book on this subject is "spread spectrum systems" by Dixon.
GPS is also based upon CDMA.
This work was first developed by Claude shannon and was top secret for many years post war.
Best Answer
To express a ratio in dB, the ratio must be unit-less, since the logarithm of the ratio must be taken, so I'm not sure I understand why you're puzzled that we use dB.
dB is often used to express unit-less ratios precisely because of the properties of logarithm.
For example, multiplication becomes addition, division becomes subtraction.
Also, since the the signal my be many orders of magnitude greater than the noise, it is more convenient to express the SNR as, say, 50dB rather than 100,000.
The phrase "the SNR is 50dB" is equivalent to "10 times the log of the ratio of the signal power to noise power equals 50."
The dB is not a dimensionful unit like a unit of length or of time, it is a dimensionless unit.
The number x is a pure number just as the number \$y = 10 \log(x) \$ is though we might say that "y is just x expressed in dB".